My kids are mixed, black and white, and they've run into some problems with others, but not as bad as I would expect in the south (I live in Atlanta).
Some kid jumped my son, age 13 then, at school two years ago and called him a "bi-racial freak" and tried to whup up on him. He (the other kid) lost his fight. I normally would have been on my son about fighting, but I let it go because it was for a good cause.
My daughter who is 14 now has trouble finding "serious" (as serious as you can get at 14) relationships at her mostly white school. I'm sure that will change when the boys mature and can look past the color of her skin and see what a wonderful person she is.
My youngest who is 10 hasn't seen too much either way on the subject. I'm sure she'll see her fair share growing up, but at this point, she's not too worried about being different.
My feelings are that if you can't except my kids, then it is your loss not mine. As long as you don't get hostile, I'll just ignore your stupidity and move on.
PH
PS - I'll be watching this thread closely for more responses from "mixed" kids and parents. I don't know that many myself, so I'll be trying to get some real insight here.