That's an excellent question. I'll answer it with a few anecdotes:Francis, I'm a black male solo traveler. I've heard that Eastern Europe might not be safe for me. What do you think?
1. In Belarus, I met a Nigerian in Minsk. He told me that every few hours police would stop him and ask him for ID and visa and papers. They held him for hours trying to get a bribe out of him - he refused and they eventually let him go.
2. I met a guy from Chad in Montenegro. He was a center-forward for the soccer team in Montenegro (and before that for a Serbian team). He said he had hardly any racism. He expected more, but said that nearly everyone treated him super well.
3. In 2010, Piran, Slovenia elected a black guy from Ghana as their mayor. He was the first black mayor in Eastern Europe. They called him "Europe's Obama."
CONCLUSION: It's a mixed bag. Most places in Eastern Europe are safe for a black guy to go, although I wouldn’t walk around alone late at night in Ukraine/Russia/Belarus, if I were you. In a group or during the day, you’ll be fine. In all the other countries, they’re quite accepting (though curious). They’re especially tolerant in the westernmost parts of Eastern Europe.
BTW, two white Americans I know got beat up by 4 Russians in Tallinn, Estonia. The Americans were being somewhat provocative, though, so they were being somewhat stupid. The point is that you don’t need to be black to get your ass kicked in Eastern Europe.
MY ADVICE: Dive fearlessly in the Balkans. Be cautious the further northeast you go (toward Belarus/Russia). You'll get MANY looks EVERYWHERE you go. Hell, I'll even stare at you if I see you in Romania, and I'm from San Francisco! However, staring is different than racism. If I go to Africa, people will stare at me in some places, but most won't bother me beyond that. Have fun!