Traditional Christianity believes that the Feast of Weeks or Pentecost is the beginning of the church or the church age. It is also believed that the church is under a new covenant as opposed to the old covenant that Moses and the Mosaic Law represented. This understanding of a new covenant comes from Jeremiah 31:31-33.
Jeremiah 31:31-33 (KJV)
Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah:
Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD:
But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
Verse 31 does indeed speak about a new covenant that God was going to make. According to this same verse, however, with whom would God make this covenant? It says that the covenant would be made with the house of Israel and with the house of Judah. In other words the covenant would be made with the whole nation of Israel. Notice that there is no entity called the "church" in this passage. Verse 32 speaks about the new covenant being different from the old covenant that the house of Israel broke. Verse 33 tells us how the new covenant would be different. It says that the Law would be put into the inward parts of the people and would be written on their hearts. The old covenant was not internal, but external, and the people could not keep the Law. With the death and resurrection of Yeshua (Jesus) the Holy Spirit was sent on Pentecost to indwell God's people. When one is born again that person receives the Holy Spirit and the Law is internalized making him or her able to keep it. Notice that this passage says nothing about the Law being changed. It is the same Law. Only its position has changed.
So how can a Gentile participate in this new covenant?
Ephesians 2:11-14 (KJV)
Wherefore remember, that ye being in time past Gentiles in the flesh, who are called Uncircumcision by that which is called the Circumcision in the flesh made by hands;
That at that time ye were without Christ, being aliens from the commonwealth of Israel, and strangers from the covenants of promise, having no hope, and without God in the world:
But now in Christ Jesus ye who sometimes were far off are made nigh by the blood of Christ.
For he is our peace, who hath made both one, and hath broken down the middle wall of partition between us;
This passage in Ephesians tells us that by the blood of Christ the wall of partition that separated Jews and Gentiles has been broken down and that Gentiles can also be born again, indwelt by the Holy Spirit, and become part of the commonwealth of Israel. Then both Jews and Gentiles can be God's people.