I'm not a US citizen nor have I ever been to the US, but here in the UK we are pretty diverse and I'm of a pretty mixed background myself (Greek and Armenian from my father's side, and Scots, Irish, English and Welsh from my mother's side).
I've never heard the term African-British, nor Hispanic British or South-East Asian British.
But I have heard the term Greek-American and Armenian-American ALL the time from that side of my family.
I think it has a lot to do with personal feelings of heritage. A lot of minorities seem to prefer the hyphenated terms maybe because they never felt any affinity to the US as a home. I don't immediately see how the could feel such an affinity towards a place/country/continent they 've never been to, but hey... it's just how they feel I guess. I don't understand it, but neither do I understand how some people like Harry Potter and Lord of the Rings.
The only reason I call myself Greek/British is because I actually grew up and lived in both countries, so I do feel that I'm part of both (just in case my dual nationality and two passports weren't confirmation enough for some...), but I would never refer to myself as "Scottish" or "Irish" just because my maternal grandparents were from there. I'll identify my ethnicity as such but not my personal feelings of cultural identity.
That's why i don't bother with ethnic titles, all they do is create more barriers and labels. Nationality is a social construct based entirely on geographical restrictions that will some day be gone. I'm a human being first, a man second and a Caucasian Greek Brit last.