Colossians 2:16-17
16 Let no one, then, judge you in eating or in drinking, or in respect of a feast, or of a new moon, or of sabbaths,17 which are a shadow of the coming things, and the body [is] of the Christ;
Others have brought up that the Scriptures saw that the believers met on the first day of the week.
In more detail, the Sabbath was celebrated to keep it holy in rememberance of God's rest, or the finished work of creation. The Old Testament as a whole, according to Christian theology (see the Epistle to the Hebrews) foreshadows and points to Christ and the New Covenant. The Sabbath is fulfilled. How, do you ask?
When Jesus died and resurrected 3 days later it was on the first day of the week that he rose. Christians have sometimes called this the "8th day" because it marks the new creation. Through baptism we are made new creatures in Christ Jesus. This present, old age is passing away and the present age has come in Christ and his Church and will come in its fullness when he returns. We no longer celebrate the old creation, but the new creation.
Sabbath is also a typology for Christ. Jesus said "come unto me all ye weary and heavy laden and I will give you rest." Christ is our new Sabbath. We no longer rest on a day, but in a person. It is a perpetual Sabbath in which the works of the Law are no longer required of us. Christ is our Sabbath and we remember it on the day of his resurrection.