How is modern day Christian morality reconciled with the morality of the Bible? Morality changes over time
Morality is not a constant; what a group of people finds to be morally acceptable or morally unacceptable evolves over time, generally because of an increased understanding or awareness, and generally this occurs (we like to think) in a more progressive, positive direction. Note: it is irrelevant whether our moral views have or have not progressed in positive direction; all that matters is that it has changed.
For example, 100 years ago (and still to some extent today), many white people (Christians included) believed that blacks were an &quot;inferior race&quot;. Before that, slavery was considered acceptable in many parts of the world. Before that, the mistreatment and subjugation of women was acceptable. Today, there are still people who think it's wrong to be homosexual, and I'm sure as with the others those views will fall more a more out of favor.
The Bible doesn't change
But as you know, the Bible is static; it's words don't change and it is never updated. The lessons it gives are supposed to be independent of the time or age you are born in. But it is quite clear from reading the Bible that some of passages espouse a morality that is no longer accepted today. How is this reconciled with today's views?
For example, imagine a father reading portions of the Bible to his son back in 1610. He comes to Leviticus 27:3-7:

Set the value of a male between the ages of twenty and sixty at fifty
shekels of silver, according to the sanctuary shekel; 4 for a
female, set her value at thirty shekels; 5 for a person between the
ages of five and twenty, set the value of a male at twenty shekels
and of a female at ten shekels; 6 for a person between one month
and five years, set the value of a male at five shekels of silver
and that of a female at three shekels of silver; 7 for a person
sixty years old or more, set the value of a male at fifteen shekels
and of a female at ten shekels.

This father would have just told his son that women are worth less than men. Back in 1610, that might have been an acceptable moral view, so he thinks nothing of it. But zoom forward to 2012: a father reads the same passage to his son. Surely he must say something about that passage, lest he wants his so to think women inferior.
Should you choose to personally re-interpret this passage as being morally acceptable, that's fine, but there are numerous other passages in the Bible which suitably demonstrate my point just as well. For example, it's generally understood in the modern world that beating our children with a rod (Proverbs 13:24, 20:30, 23:13-14) is a little... &quot;Old Testament&quot;, to say the least. Even if you still think beating your children is okay, I doubt you think it's okay to kill them if they talk back (Leviticus 20:9, Deuteronomy 21:18-21, Mark 7:9-13, Matthew 15:4-7, Exodus 21:17). I doubt most of you still support slavery, which is supported in both the Old and the New Testaments (Matthew 10:24/24:45-46, Ephesians 6:5, 1 Timothy 6:1-4), or find it okay to give your daughter as a slave (&quot;maidservant&quot; in some of the prettier editions). Let's be real here, there are passages in the Bible which contain morally questionable messages; this isn't a bold claim.
Since most Christians seem to be normal, decent people, what's going on here?
I honestly just don't know because my own parents—while religious—never even tried to explain the Bible to me. In fact, my parents for the most part completely dismissed the Bible, and engaged in Christianity in a more organic, &quot;everyday positive lessons&quot; manner. But for those who actually take the time to sit down and read the Bible, whether in Church or at home, what do you do when you come across passages such as these? Do you simply pass them off as historical relics of a long-expired moral past, or do you try to justify it as some sort of metaphor that accords with our beliefs today?
Biblical Christianity has not changed in its definition of morality and will never change, since it is based on the Bible, which, as you pointed out, does not change.

However, there is an  important distinction between Gentile believers and the Old Testament Jewish Law.  The Law was a specific covenant with a specific people for a specific time, but it was a shadow of things to come.

While the New Testament did not usher in a differing morality, the ceremonial law was no longer in effect, and it was never applicable to Gentiles.

According to Biblical Christianity, adultery, theft, lying, deceit, slander, gossip, malice, lust, profanity, pride and homosexuality have always been immoral and will always be immoral.

The passage regarding women, as X-Zero pointed out, has to do with the value that each gender could contribute in an agrarian society.  Having grown up on a farm myself and being 6'4" 250+ lbs., I have no problem stating today that men still are much more suited to hard physical labor and are, consequently, more productive.  In fact, my perspective is that I as a man should perform very laborious tasks (and very dirty task as well), so that women don't have to.  Equality does not mean equivalence.  Man != Woman.  So, this is not an antiquated passage.  The principle still holds today.  If you could hire me or a typical 5'6" 1xx lb. woman to work in your fields, you would likely offer me more money, since my capacity and production in your fields would be much greater.  Yet, if you could hire me or a woman for a job that is not labor-intensive, then you might offer the woman more if she is more qualified, more experienced, or just does a better job.   So, this example is really a moot point.