The Bible seems to allow polygamy. Why doesn't the church? In Exodus 21:10 and Deuteronomy 21:15-16 (among many other passages), it talks about polygamy as if it were acceptable.

Indeed, there are several holy men (Abraham and Solomon come to mind offhand) that had multiple wives. 

It seems pretty clear that God doesn't mind polygamy and that it is allowed.

Why is there a modern-day restriction on such practices?

Edit
As has been pointed out, I'm using "polygamy" in the modern/popular sense (such as "polygamy camps" of the FLDS group).  Truly, the actual word is "polygyny", which is a man marrying multiple wives.  Other forms of polygamy (polyandry and such) seem pretty clearly against God's word.

Edit 2
Per a comment, I'm seeking any answer from any denomination/doctrine that does not support polygamy.  Although there are a few denominations that do, I'm not interested in their history so much.
Note: Throughout, I use the word &quot;polygamy&quot; in place of &quot;polygyny&quot;, even though I explicitly mean to polygyny.

In short:
Polygamy is a sin because it goes against the law.  The law is in place because it is a carryover from the paganistic societies of ancient Rome.  Preventing polygamy was not a biblical concept, but one that came after Jesus, after authorship of the bible, and after the genesis of Christianity.
Why Polygamy is a sin
The reason that polygamy is no longer allowed by the church is because it's no longer allowed by society and is, therefore, against the law.
We Christians are told to uphold the law and subject ourselves to the law
Romans 13:1-2 (NIV)

1 Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God. 2 Consequently, whoever rebels against the authority is rebelling against what God has instituted, and those who do so will bring judgment on themselves.

Because of this, polygamy is a sin if (and only if) it breaks the law.  Similarly, it is a sin in the same manner and degree that breaking a speed limit is a sin: it's something not forbidden by God and solely forbidden by the law.
Why it's illegal
Polygamy was not uncommon among the Jews in biblical times.  It wasn't until the the Romans came in that polygamy became outlawed.  Even then, it was still allowed; Josephus made notes explaining that polygamy was permitted to Herod because it was permitted by Jewish custom.
Polygamy modelled by Jesus's parable
We can see that polygamy was commonly accepted in the New Testament from the Parable of the Ten Virgins (Matthew 25:1-13).  In this parable, Jesus tells of a bridegroom that is preparing to marry ten women.  This parable was used by Jesus to describes himself.  The polygamous bridegroom in the parable is a reference to Jesus.
Fall of polygamy
It was, in fact, the Greek and Roman rules against polygamy that spread (along with the Grecian and Roman empires) and became the culture, forcing out the practice of polygamy.
Monogomy and polygyny by Walter Scheidel, Stanford University (2009)

Thus, even though Greeks and Romans need not have been the first cultures to prescribe monogamy, these are the earliest securely attested cases and, moreover, established a paradigm for subsequent periods that eventually attained global dominance.
...
The true historical significance of Greco-Roman [monogamy] may well lie in its impact on the
Christian tradition.

It was these two huge historical forces that established monogamy in our society--not something born of God, but something born of Pagan societies.
Modern Society and Law
Christianity tied itself to the banner of the Roman empire 300 years after Christ (when Constantine established what would become the Roman Catholic church).  This organization spread Christianity by using the power and authority of the Roman empire.  It was this Roman concept of monogamy (predating Christianity) that carried over from its paganistic roots into the Christian society.  It was at that time (and not before) that Christianity became intertwined with the idea of monogamy.
This pagan idea of monogamy from Rome slowly infused into the Christian culture (source); and it was because association with Christianity (and the Roman Catholic Church) that the paganistic, Roman practice remains today.
This Roman practice has tangled itself into the Christian culture (via the early Roman Catholic church) and therefore all &quot;Christian&quot; nations since (founded under Western culture) have adopted the laws prohibiting polygamy from their outset.
Because of this, polygamy is illegal in most parts of the world.  Because it's illegal, it is a sin.
Catholicism Today
It should be noted that Catholicism today still carries over this idea from ancient Rome.  In the Catechism of the Catholic Church, it states &quot;However polygamy is not in accord with the moral law.&quot;
Because of this, polygamy is solidly outlawed within Catholicism.  Granted, if you read the last section, this should not surprise anyone.

Answering Aguments Against Polygamy
God tolerated the sin for the time being
Malachi 3:6 says The Lord does not change.
To presume that he would accept something during one time in history that he detests at a different point is crazy--heretical even.  It goes against the nature of God (both in the Old Testament and New)
The bible shows it only causes trouble in marriage
This is really a weak argument.  The problem is that the Bible only tells us stories that are either edifying or interesting.  Yeah, it's going to talk about Abraham's difficulty with his wife and taking up a handmaiden as a wife.  But that story is both interesting and edifying.  The lesson here isn't against polygamy, but rather that we shouldn't marry a woman unless we marry her properly and with the right intentions.
Jacob's wives competed.  But the story there isn't to avoid polygamy, but to focus on God and make your marriages holy and godly.  If your marriages are focused on God instead of on selfish ambition, this won't happen!
Etc.
Each argument that says &quot;polygamy is wrong because it leads to dischord&quot; can also be used to say &quot;marriage is wrong&quot; for the same reasons.  Those are the marriages not focused on God--that is the source of their dischord.
On the contrary, if we look at the life of David, we see that he was a man highly loved by God and yet had multiple wives.  Some, however, use this very example to show how polygamy is wrong because his children fought (with each other and their parents).  However, can we blame this on David or his wives?  Absolutely not.
God only created one Eve for Adam
Note that this was written by Moses, who had (at least) two wives
This is also a weak argument.  When God created Eve, he created her as a helper.  Whether or not he intended for this to be a marriage situation is of secondary concern and something that we cannot know 1 2.
On the contrary, though, God has given us a pattern to follow for marriage:
Ephesians 5:22-25

22 Wives, submit yourselves to your own husbands as you do to the Lord. 23 For the husband is the head of the wife as Christ is the head of the church, his body, of which he is the Savior. 24 Now as the church submits to Christ, so also wives should submit to their husbands in everything.
25 Husbands, love your wives, just as Christ loved the church and gave himself up for her

We are to model our marriages after the the relationship that Jesus has with the church, not after the helper relationship that Adam had with Eve.
Furthermore, in the Parable of the Ten Virgins (Matthew 25:1-13), Jesus describes himself using an analogy of a bridegroom that is preparing to marry ten women.  Why would Jesus really compare himself to a polygamist if it were detestable to him?
&quot;Elders should be men of one wife&quot;
Many use Titus 1:6 and 1 Timothy 3:2,12 against polygamy.  They say that these verses talk about how deacons, overseers, and elders should be men of &quot;one wife&quot;.
Point 1:  Simply because elders should be a &quot;man of one wife&quot;, does not prevent polygamy among everyone else in the church.
Point 2: translation of this &quot;one wife&quot; is not as cut-and-dry as it may seem
To fully understand point 2, I want to compare these three passages to 1 Timothy 5:9:

A widow is to be put on the list only if she is not less than sixty years old, having been the wife of one man

The greek for the word &quot;one&quot; here is heis. From Strongs Concordance 1520, this word is translated as one, single.
However, if we look at Titus 1:6 and 1 Timothy 3:2,12, the word we see for &quot;one&quot; is mias.  This is the same word used in Matthew 28:1 for &quot;first&quot;.
While it's not definitive that mias should be translated as &quot;first&quot; instead of &quot;one&quot;, it definitely shows that &quot;a man of one wife&quot; can have multiple translations.
In fact, let's look at the NIV version of this:
Titus 1:6 (NIV)

An elder must be blameless, faithful to his wife, a man whose children believe and are not open to the charge of being wild and disobedient.

&quot;faithful to his wife&quot;?  Why did they translate it that way?  Is it because they're trying to &quot;please the polygamy crowd&quot;?  Absolutely not!
The purpose of this passage is to indicate that the elders should be above reproach, faithful to his wife.  If you accept the mias as &quot;first&quot;, it changes the translation of this passage to &quot;man of his first wife&quot;, which means someone who's never been divorced!
That is the purpose of this passage: not to exclude polygamist, but to find men who should be held up as models for the community.  Men who are faithful and can maintain a marriage faithfully.
Two become one flesh
Note that this was written by Moses, who had (at least) two wives
Genesis 2:24

That is why a man leaves his father and mother and is united to his wife, and they become one flesh.

There are several passages that show that a man becomes &quot;one flesh&quot; with a woman regardless if it is his wife or a prostitute.
The idea of &quot;one flesh&quot; does not mean that the man is exclusive to that woman.  It means that they join themselves as if married.  This is true regardless of who the man does this with.
This is also clear in Ephisians 6:16-17 (NIV)

16 Do you not know that he who unites himself with a prostitute is one with her in body? For it is said, “The two will become one flesh.” 17 But whoever is united with the Lord is one with him in spirit.

It is clear that many believers can unite with the Lord as many prostitutes can unite with one man.
Furthermore, in John Skinner's Genesis: International Critical Commentary (T &amp; T Clarke, Edinburgh, 1930) p. 70, Skinner notes that the word for &quot;flesh&quot; here is synonymous for &quot;clan&quot; or &quot;family group&quot;--both in the Hebrew and Arabic.  This word for &quot;flesh&quot;, in fact, is the same word used here in Leviticus 25:49

An uncle or a cousin or any blood relative in their clan may redeem them. Or if they prosper, they may redeem themselves.

Clearly, there is no restriction of one man becoming an exclusive single flesh with a woman.
Other Arguments
There are many other arguments that can be heard and answered at BiblicalPolygamy.com