What does it really mean that we are no longer under the Law but under grace? Is sinful behaviour excused, since we are no longer under Law, but under grace?
To be under something means to be under its power. We are no longer under the power of the Law, but we are under the power of grace. The only power the Law had over men's lives was the power to condemn. But what power does God's grace have in our lives? God's grace in our lives empowers us to walk in God's ways. Therefore God's grace doesn't assume we'll continue to live lawless lives - instead, God's grace empowers us to walk in love, for God is love - and love is the fulfilling of the Law.
The terms of the New Covenant state that God will put His laws in our minds and write them in our hearts, and cause us to walk in His ways - evidently then, God still has laws, even after the annulling of the Old Covenant - or else there would be no laws for Him to write in our hearts. The difference in the New Covenant is that this time God Himself causes us to walk in His ways because He has given us a new heart and He has written His laws in our heart - that's God's grace.