How Does One Keep God's Laws?

There is an interest in some places about the need to keep the Ten Commandments. This is understandable in that the current teaching of "grace" definitely leaves the impression that there are no laws governing Christians, except possibly "the law of love," which is no law at all in that it is based on our feelings.
By the "law" the reference is to the Ten Commandments. The remainder of the Law of Moses is not emphasized as much, such as the dietary laws for example.
The question is: Are Christians under the Ten Commandments, or are they not? And if not, what law are they under, if any?
This is an important issue in our day, in that because of the defining of Divine grace as a way of having fellowship with God while we continue to yield to the sinful nature, sin abounds in the Christian churches. In fact, the idea of conviction of sin sometimes is viewed as being a relic of the dark ages.
The issue of whether Christians are free to sin is of double importance because we now are entering the spiritual fulfillment of the Jewish Day of Atonement, in which the Holy Spirit is pointing out the sins in us and giving us the strength to confess them and renounce the practice of them.
Perhaps the first concept we must understand is that the Ten Commandments are part of the Law of Moses and have no authority over believers in Christ who count themselves as dead with Christ on the cross, and raised with Him to walk in the newness of resurrection life.
Do you not know, brothers—for I am speaking to men who know the law—that the law has authority over a man only as long as he lives? (Romans 7:1)