In the current age Christians are the spiritual Israel. Hence, the Ten Commandments would still apply to us just as they did to physical Israel.
This point alludes to Romans 2:23-29: "You who make your boast in the law, do you dishonor God through breaking the law? For "the name of God is blasphemed among the Gentiles because of you," as it is written. For circumcision is indeed profitable if you keep the law; but if you are a breaker of the law, your circumcision has become uncircumcision. Therefore, if an uncircumcised man keeps the righteous requirements of the law, will not his uncircumcision be counted as circumcision? And will not the physically uncircumcised, if he fulfills the law, judge you who, even with your written code and circumcision, are a transgressor of the law? For he is not a Jew who is one outwardly, nor is circumcision that which is outward in the flesh; but he is a Jew who is one inwardly; and circumcision is that of the heart, in the Spirit, not in the letter; whose praise is not from men but from God" (Romans 2:23-29). Paul's point was that Israel claimed to keep the Law, but failed to do so. Hence, they lost the right to be called God's chosen people. This point is further discussed in Romans 9 and 11. True Jews (chosen people of God) are not those who keep the Law outwardly (i.e. being circumcised in the flesh) but those who keep it inwardly.
Paul's point is that the nation of God's people would no longer be based on blood lines or physical markings. "But it is not that the word of God has taken no effect. For they are not all Israel who are of Israel, nor are they all children because they are the seed of Abraham; but, "In Isaac your seed shall be called." That is, those who are the children of the flesh, these are not the children of God; but the children of the promise are counted as the seed" (Romans 9:6-8). Hence Peter states of Christians, "But you are a chosen generation, a royal priesthood, a holy nation, His own special people, that you may proclaim the praises of Him who called you out of darkness into His marvelous light; who once were not a people but are now the people of God, who had not obtained mercy but now have obtained mercy" (I Peter 2:9-10).
However, notice that Paul compares the circumcision (the Jews) with the uncircumcision (the Gentiles). Circumcision was a requirement of the Old Law. "You shall be circumcised in the flesh of your foreskins, and it shall be a sign of the covenant between Me and you" (Genesis 17:11). "And on the eighth day the flesh of his foreskin shall be circumcised" (Leviticus 12:3). Yet Paul stated that the uncircumcised in the flesh would become the true Jew in the Christian age. How can this be since circumcision was a sign of the covenant given to Abraham and continued through Moses? The Law is quite clear that an uncircumcised male could not be a part of the covenant. "And the uncircumcised male child, who is not circumcised in the flesh of his foreskin, that person shall be cut off from his people; he has broken My covenant" (Genesis 17:14). This could only happen if the law's requirements had changed. "But now we have been delivered from the law, having died to what we were held by, so that we should serve in the newness of the Spirit and not in the oldness of the letter" (Romans 7:6). Please take note that Paul gives an illustration from the Law in the very next verse by citing the tenth commandment -- the law forbidding coveting.
What happened to circumcision? It was replaced by baptism. "In Him you were also circumcised with the circumcision made without hands, by putting off the body of the sins of the flesh, by the circumcision of Christ, buried with Him in baptism, in which you also were raised with Him through faith in the working of God, who raised Him from the dead. And you, being dead in your trespasses and the uncircumcision of your flesh, He has made alive together with Him, having forgiven you all trespasses, having wiped out the handwriting of requirements that was against us, which was contrary to us. And He has taken it out of the way, having nailed it to the cross" (Colossians 2:11-14).
Therefore, the changing of physical Israel to a spiritual Israel does not imply that the laws given to physical Israel were transferred to spiritual Israel. The keeping of the Sabbath is another illustration of a difference between the old covenant and the new covenant. "So let no one judge you in food or in drink, or regarding a festival or a new moon or sabbaths, which are a shadow of things to come, but the substance is of Christ" (Colossians 2:16-17). The keeping of the Sabbath was only a shadow of a later reality in Christ, just as circumcision was a shadow of baptism.
Now, this does not imply that Christians do not have similar laws as the Israelites. After all both the Old Testament and the New Testament originate from the same and only God. See "Ten Commandments" in the Scripture Index for a list of verses.