When I was a kid I taught myself to ride a bike without training wheels. After this came up in conversation with a few people recently, it appears that some people know about this method and some people do not. Here it goes:

Remove one of the training wheels. Ride, and if you start to wobble lean to the side with the training wheel for support. Once you've become stable, try to center your balance again off of the training wheel. It took me about a half hour of this when I was 6 or so before I didn't need the other wheel.