Does it say anywhere in the Bible that you should not date out of your race?

In 2 Corinthians 6:14 God instructs us not to marry an unbeliever if we are a follower of Christ. His concern is not ethnic heritage or the color of our skins, as He is not prejudice against such things. His concern is always the condition of our hearts.

That is why in the Old Testament there are times when God emphasized He did not want His people Israel to intermarry with surrounding nations:

"Furthermore, you shall not intermarry with them; you shall not give your daughters to their sons, nor shall you take their daughters for your sons. 4"For they will turn your sons away from following Me to serve other gods; then the anger of the LORD will be kindled against you and He will quickly destroy you. 5"But thus you shall do to them: you shall tear down their altars, and smash their sacred pillars, and hew down their Asherim, and burn their graven images with fire." (Deuteronomy 7:3-5)

These surrounding nations were idol worshipers, not followers of God. They would lead His people astray and corrupt His message through the Israelites to the world. This same them is seen in Nehemiah 13:23-30.

God's command against inter-racial marriage was about preserving a unique group of people with whom He was establishing a nation and carrying out a very special plan to bless the world through them. Nowhere in Scripture does it say that this was a universal edict intended for all people in all cultures and time periods.

To the contrary, in many different places in Scripture God addresses the responsibility of His people to treat the "foreigner" or "stranger" or "alien among you" with respect and compassion. Including them even in the most sacred of activities, as long as they were obedient followers of God. Using your concordance or a google search should bring up several of them.

If one is concerned about choosing a marriage partner that is pleasing to God, the important first questions must be, "Have I prayed and asked God to lead me in this decision? Does this person profess Christ as his/her savior? Is this person seeking to grow in godliness? Is this person seeking to grow in obedience to God? And are these things true of me as well?"