My Name is Not Amy Tan

When I write, I always hesitate putting in my character's ethnicity.  Sometimes, I've specifically imagined my character being white, Asian, etc., but I never include that information in the story.  I tend to keep the hair and eye color pretty generic; they vary between different shades of brown. They never have a last name.

I am American-born Chinese, and though I've never specifically had someone tell me this, I feel like if I do mention a character's ethnicity in my story, then its automatically about race.