You've never tried talking to a Scotsman I suppose? Anyone that thinks the British or Irish accent is 'heavy,' wait until you speak to a Scot... there is a DEFINITE difference, almost to the point of it being a different language.
I will admit, however, that I do have a difficult time differentiating between the subtle differences in facial features between say Chinese, Korean and Japanese, but I never make an assumption someone is descended (or is, if they are visiting the US) from one or the other, or anything else.
After speaking with an American of English/Japanese descent, I've learned people take BIG offense when you say shit like 'What are you?' Mike's reply is always 'American, what are you?' People don't like his ignorant reply either.

He also replies with about the same when people say 'Where are you from?'