One of the things that makes it so difficult to think about how the law applies, or if the law applies, to New Covenant believers is the inconsistent use of the word "law" (greek 'nomos') in the New Testament, especially in Paul. For example, Paul may use the word "law" to refer to the entire Pentateuch (Romans 3:21). In the same verse, however, Paul uses the word "law" to meaning something like "works of the law". In Romans 3:27 he uses to word to mean something like "principle". In still other places it seems as though Paul uses the word "law" with special reference to the Judaizers "law keeping" as a means to justification. It may be possible to add more nuances, but you get the idea.
So, when asking if the law applies to New Covenant Believers, we must be careful that we know what sense of the law we are referring to. Certainly we would all want to agree that much of the Pentateuch applies to us. So we aren't dead to the law in that sense. Moreover, I don't think anyone would want to argue that we are under the law in the sense of the Judaizers law-keeping which leads to justification (that was never the intention of the law when given at Mt. Sinai, how could it be now?).
Ok, that aside, let me state my position: I do believe that we, as Christians, are obligated to keep the law. Now that needs a lot of clarification, so stick with me (not just in this post, but over the next few).
I think it is important to realize that there is an element of law in each of the biblical covenants. Law was not introduced in the Mosaic era, though law certainly does play a much more prominent role in the Mosaic covenant. Actually, I don't even know if saying like that is as accurate as I would like to be. Let me try again: the external codification of law plays a much more prominent role in the Mosaic covenant than in the other Bible covenants (that's better, but still needs work).
You can certainly find law in the pre-fall covenant God imposed upon Adam (sometimes referred to as the "covenant of works" or the "covenant of creation". In Genesis 1&2 you see that man was given responsibilities unique to his status as an image bearer. He was to exercise dominion over creation, and he was to multiply and fill the earth. In addition, he was given the specific command - not to eat from the tree in the center of the Garden.
The Noahic Covenant contains commands as well. The covenant relationship begins with the command to build the ark. Again, at the inauguration of the covenant in Gen 9, Noah receives the command to "be fruitful and multiply", the prohibition against eating food with its blood still in it, as well as the decree of God's will regarding murderers (v.5-6).
In the Abrahamic Covenant, law is important as well. At the outset of the covenant relationship, Abraham is required to leave his land and family and set out the place God would show him. In addition, as the covenant with Abraham is more fully established, the seal of the covenant, circumcision, becomes a mandatory rite for those who would enjoy the blessings of the Abrahamic Covenant. Stern warnings were issued against neglecting this sign/seal, and severe penalties were to be actuated against those how spurned this covenant rite (see a rather startling instance of this in the life of Moses recorded in Exodus 4).
In the Davidic Covenant, again law is important (though I'll say it again, not as prominent as in the Mosaic covenant). God speaks of discipline and correction for law breaking as he makes this Covenant with David (2 Samuel 7:14). Certainly that only makes sense in terms of law and law breaking. Moreover, the role of the Mosaic Covenant and its laws is clearly articulated in Davids words to Solomon in 1 Kings 2:1-4. Further, the history of Israel is a often sad reminder that law breaking has dire consequences to the people of God.
That brings us, finally, to the New Covenant. Again, I believe the law plays a important role in the New Covenant (I'll deal more specifically with some of Paul's comments regarding the law at a later date). It is extremely important to note that it is not freedom from the law that the OT prophets look for, but instead freedom to keep the law. In Jeremiah 31, God promises that when he establishes his new covenant with his people he will "put my law within them, and I will write it on their hearts." (31:33, ESV). Moreover, that Jesus makes demands of his followers can hardly be denied (I like the title of Piper's newest book, "What Jesus Demands of the World").
For now, I hope I've established that the principle of law transcends the Mosaic Covenant. We'll have to give more detailed consideration as to how the law as administered under the Mosaic Covenant applies to us today in a subsequent post.