Without diminishing your personal experience, I should point out that kids can be pretty harsh and that almost all of us are picked on in one way or another, for a variety of reasons.
You know in 'Revenge of the Nerds' where at the end, the Nerds make a call to everyone who felt slighted and diminished to come out and stand against the 'Alpha Betas' (basically the cool/athletic alpha-bros?) - and everyone came out!
95% of people have either something that stands out that could enable them to be picked on, or, give them some kind of anxiety one way or another.
So, the adult world is very different.
I suggest there is a very low likelihood that you'll encounter acute problems (though maybe kids at school), issues would be more systematic than acute, i.e. the more local people have tighter social circles - much like Europe is very 'open, tolerant, progressive' both intellectually and in terms of policy, and yet at the same time 'very, very closed' socially. Even a white person, born and raised in Paris, utterly fluent and indistinguishable from anyone else ... but with a Polish last name ... might have some difficulty integrating outside of Paris. It's not like the French 'hate Poles'. It's just a kind of ethnocentrism that is fairly common in most places.
Other than maybe your kids having to deal issues perhaps a little more often than the kids next to them in class ... I wouldn't ever be apprehensive about living most places in the US at all.