Updated October 13
I finished the Oswald Dance exercise, as suggested by John K's blog. It's not so great, but I feel like I learned a lot.
Here's a link to the Flash animation):
Oswald Dance Excerpt
I put the ears on a separate layer, so that I could reuse the drawings for the pivot cycle. The body pivots in six frames each direction. When it pivots back the other way, the same six body shapes are used in reverse. However, the ears have different shapes because they follow the movement of the head. So I'm reusing the same six body drawings, with twelve different ear drawings.
The mouth is also on a separate layer, again so I can reuse the body drawings.
I drew the body with pencil and paper. I drew the ears and mouth in Flash with a tablet.