I see nothing wrong with stating a character's race if it is unexpected. If my story takes place in Japan, and one of the employees of a major business was white, I'd say the character is white, and not pussyfoot around it by giving her "white features" or making her do stereotypically white things.
Again, it's set against a default race. By default, the readers might expect everyone to be Japanese (which is probably true, since Japan is a homogenous society). Like Claudia said, if you set the story in a homogenous society like Sweden or Japan, then go on and assume. Otherwise, if you're writing about modern-day NYC, then no, it's not good to assume everyone is white or whatever.
Still, does it MATTER? That's a question you need to ask, per your story, characterization, plot, etc. Do you have to mention the race? What purpose does it serve when a character say, "Mr. Andrews is a white guy"? I think just having Mr. Andrews to be among all the Japanese characters would be enough for us to know that, well, Mr. Andrews may not be Japanese (unless he is, then we might want to offset the false expectation): Mr. Andrews is of Japanese descent but was adopted by Caucasian parents.
Like my friend Amy. She's Korean but her maiden name is Anderson. Often she has to explain to people why. On the other hand, many people really don't give a crap. I once went on a flight using my colleague's ticket (that was before 9/11), and his name was Patrick Nicholas. No one ever asked me, "why the heck did an Asian guy like you have a name Nicholas?" It is because, it doesn't matter.
When writers tapdance around words like "black" it makes me think that minorities make them uncomfortable and therefore they have no business putting token characters in their novels.
There's some truth in this. I don't even think about it when I write about people of different races. Most often I don't find the need to have to explain or describe their races -- they're just people. I have no need to say "X is white, and Y is black, and Z is Latino." I think when you think about being PC or being "not offensive," there's probably something deep-rooted as far as races are concerned....