4 Likewise, my brothers, you also have died to the law through the body of Christ, so that you may belong to another, to him who has been raised from the dead, in order that we may bear fruit for God. 5 For while we were living in the flesh, our sinful passions, aroused by the law, were at work in our members to bear fruit for death. 6 But now we are released from the law, having died to that which held us captive, so that we serve in the new way of the Spirit and not in the old way of the written code. 
(Romans 7:4-6 (ESV)
So we have died to the law, we are no longer bound by the law. We died through the body of Christ to which we were incorporated, embodied in baptism, and that body died fulfilling every aspect of the law. So we no longer serve the law.
It has to be noted when dealing with this passage that it is the law that arouses sinful passions. Far from doing what we seem to think the law does, the law does the opposite. We think the law should put down sinful passions. This is what drives much of what passes for Christian preaching these days. That if we just preach enough law, our members will be adequately warned and they will avoid the sin in question. But the law doesn’t do that. The law actually arouses the sinful passions within us, the law makes us more inclined to sin. It increases sin.
But we have been released from the law, we serve in the newness of the Spirit, which would be a better manner of translating this as “way” is a word that the translators seemed to have added for clarity. But this isn’t a matter of a way of serving, it is a matter of a new state of being from which we serve. It is harkening back to the “newness of life” in Rom. 6. And this is contrasted to “and not in the old written code.” Here Paul is talking about the law as it is written in the Old Testament, and especially about the law as it was written in stone by God himself. This means it is no longer a condition that we have to follow if we want to earn heaven. We don’t get to heaven by the law. The Bible is not in this sense “basic instructions before leaving earth” (I really hate American evangelical cliches). It isn’t that we are “freer” in regards to the law, that we get to basically tweak it this way and that to suit our own needs, and use it the way most of us use speed limit signs as a guideline…. The law doesn’t work that way. The law kills that way. And the law is an all or nothing proposition. Either we are beholden to it all or we are not. But now we are free from it. Now we serve God in the newness of the Spirit as we walk in the newness of life. Now the law has nothing to do with our salvation because our salvation has been given to us freely apart from it. Now the law can never be anything more in our life, but an expression of our love for neighbor. It’s no longer about us, it’s about our neighbor. Or to put it this way, following the law can’t save us, but us following the law might save our neighbor, in that in showing love for our neighbor they come to want to hear us when we speak the gospel of Christ’s fulfilment of the law for us.