Okay, so then here's my dilemma: I have never yet found any kind of personality characteristic that can define a race of people. So obviously, I can't indicate the color of a character's skin with a description of his or her actions, beliefs or personality.
Do you feel hesitant to mention a white character's blonde hair, the fact that she blushes when a boy talks to her or other features? You're right that it's not personality that determines their race, but there can be food, dialogue patterns, reactions of other people, names, analogies and other means to give the reader a clue. It's not necessarily a stereotype if it rings true to the character. Even if you have one minority character in an all white environment and that character was born and raised alongside of the other characters, I'd think there would be some time that the character or someone else comments on that. Or maybe it is just part of the character's inner dialogue. Maybe she worries about who to date or if the real world is as colorblind as her experience has been.
Or if you are creating a world where race isn't an issue, then you can include that as part of your world building. Do something to let the reader in on the fact that the people are all different but it doesn't matter in this world.
In one book, I compare white tissue paper that a character's using to wrap up an object to the skin of the old woman who gave the object to her. It was relevant that the old woman was white and the passing of the object created an image of the old white hand handing something to the young dark hand without slamming the reader over the head. Other times, the character's own perception of him/herself can help. Most people (especially young ones) obsess over their looks. Depending on your point of view, you can sneak things in while helping the reader understand the character's self-perception.
Or do I have to I've never, ever lived in a town or city that wasn't racially diverse, so I always assume that the towns in the books I read are too...
That's great that you've had that experience. If you ever doubt that race still matters to a lot of people, just scroll down the comment section in any story about a person of color on the Yahoo news feed and regardless of whether it is about race or the race of the person is even relevant, check out the comments. Also, realize there are parts of the country that are extremely segregated. Not necessarily because of beliefs, but jobs, opportunities, the size of the town, the part of the country--many other factors can influence whether an area is diverse or not. We are obligated to explain our setting, situation and assumptions to our readers without them realizing that's what we are doing.
On a side note: I thought my parents were completely not racist until I brought home a black boyfriend. They came to love him (getting to know people tends to do that) and he was completely wonderful about it. I was shocked, hurt, horrified. My boyfriend (now husband) was not. He expected it. I did not. Sometimes, it is more about whether we are paying attention than it is about whether it exists. My parents didn't expect that reaction in themselves either. Keep feeling the way you do because the more people who feel that way, the better things will get-- and it's a lot better. But when you write, take off all blinders and be harshly honest with yourself and your characters and see if it helps. Make some mistakes and fix them. You'll figure it out.