Are our consciences trustworthy? In Romans there are a couple verses that indicate that the law has not only been given in written form through the ten commandments, but written "on our hearts." It also says that this law will bear witness against us. 


  Romans 2:14-16 (ESV)
  For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them on that day when, according to my gospel, God judges the secrets of men by Christ Jesus.


First, is this our conscience? Secondly, is this "law written on our hearts" corrupted by our sin natures or pure and trustworthy? In the next chapter goes on to say that there is no one who does right or understands. Particularly in light of those verses and the doctrines of original sin and total depravity, what does should Christians believe about the purity and trustworthiness of our consciences?
Yes
The word conscience comes from the Latin conscire, a compound of con (with) and
scire (to know), meaning “to know together with,” “joint knowledge with another.”
Your conscience is your ability to know right and wrong in connection with the laws of God that have been written upon your heart at your new birth of the Holy Spirit. Heb. 8:10-11
Your conscience is the result of God's breath and life within you.

Genesis 2:7 (NIV)
7 Then the LORD God formed a man[a] from the dust of the ground and
breathed into his nostrils the breath of life, and the man became a
living being.
Job 32:8 (NIV)
8 But it is the spirit in a person,     the breath of the Almighty,
that gives them understanding.
Proverbs 20:27  (NIV)
27 The human spirit is the lamp of the LORD     that sheds light on
one’s inmost being.

The prophets relied on their consciences

Romans 2:14-15  (NIV)
14 (Indeed, when Gentiles, who do not have the law, do by nature
things required by the law, they are a law for themselves, even though
they do not have the law. 15 They show that the requirements of the
law are written on their hearts, their consciences also bearing
witness, and their thoughts sometimes accusing them and at other times
even defending them.)
Romans 9:1 NIV
I speak the truth in Christ—I am not lying, my conscience confirms it
through the Holy Spirit—
2 Corinthians 1:12
Now this is our boast: Our conscience testifies that we have conducted
ourselves in the world, and especially in our relations with you, with
integrity and godly sincerity. We have done so, relying not on worldly
wisdom but on God’s grace.

Weak consciences
Your conscience can become weak and defiled if you do not keep it clean.

1 Corinthians 8:7
But not everyone possesses this knowledge. Some people are still so
accustomed to idols that when they eat sacrificial food they think of
it as having been sacrificed to a god, and since their conscience is
weak, it is defiled.
Titus 1:15
To the pure, all things are pure, but to those who are corrupted and
do not believe, nothing is pure. In fact, both their minds and
consciences are corrupted.

Here's a great article if you want to get into some really deep reading regarding your conscience.