I have never heard it explicitly preached that "Jesus is our Sabbath". However there is certainly Christian teaching that God promises a Sabbath rest for his people, apart from the Seventh Day sabbath (sabbaths by the way referred to all kinds of rest and special periods, not just the seventh day). An important passage in this teaching is Hebrews 4:1-13:
For if Joshua had given them rest, God would not have spoken later about another day. There remains, then, a Sabbath-rest for the people of God; for anyone who enters God’s rest also rests from their works, just as God did from his. Let us, therefore, make every effort to enter that rest, so that no one will perish by following their example of disobedience.
That passage doesn't directly address the question of sabbath-keeping on Earth. However other Bible passages do.
Therefore do not let anyone judge you by what you eat or drink, or with regard to a religious festival, a New Moon celebration or a Sabbath day. These are a shadow of the things that were to come; the reality, however, is found in Christ.
And Romans 14:1-14:
Who are you to judge someone else’s servant? To their own master, servants stand or fall. And they will stand, for the Lord is able to make them stand. One person considers one day more sacred than another; another considers every day alike. Each of them should be fully convinced in their own mind. Whoever regards one day as special does so to the Lord. Whoever eats meat does so to the Lord, for they give thanks to God; and whoever abstains does so to the Lord and gives thanks to God.
There isn't universal agreement that these passages mean we do not need to keep the Sabbath, but they are widely cited verses. The Matthew 12 passage is also often combined with the nearby passage "The Sabbath was made for man, not man for the Sabbath" to indicate that the Sabbath is not of ultimate importance.