As an african-american male in tech, the advice I'd give to everyone is:
1. Have empathy -- this is just general advice, would apply anywhere. Remember that meritocracy is great in theory, but hard to achieve in practice because of the various (and differing realities) things that people have to live with (whatever gender/race/etc).
2. Try to avoid racially charged terms like "whitey".
3. Try to shed preconceptions of what another person of a different gender/race/ethnicity/whatever likes, or does, or prefers. Always ask (this also lets you get to know the person, which is nice), and clarify, even if it seems silly.
I say try because I make a conscious effort to do these things daily. I've done pretty well for myself, but there are lots others that are struggling, my race or not.
I don't think you're guilty of anything just by being born of a race that currently has an advantage. Parents also work tirelessly to give their children the best possible chance in life, some parents succeed, others don't. Even if it wasn't race based, it'd be economically based.
I'm personally of the belief that the only time race relations will really be fine is when everyone who was involved or remembers the bad times is dead. Then the question is whether we're doomed to repeat it or not.