Some do, some don't. It's up to personal preference and taste, just like other things.
But are you happy if a Japanese girl approaches you just because you're caucasian? Doesn't it sound as if anyone will do? The only the value and importance of you for her is being caucasian?!
Likewise, if you approach a Japanese girl not because what she is (i.e. you find her cute, charming, friendly, etc.) but just because she is Japanese, she would not be best pleased because it looks as if anyone will do for you as far as the person is Japanese.
If you like a girl and she happens to be Japanese, then it's a different story.
It will be a good idea to follow what Obeika suggested - he's giving very a very good advice.