Woo hoo! I've tried 5 or 6 different slideshow options and I finally found one that does exactly what I want.
This module is easy to install, easy to configure, is centering itself in the side rail, has a nice transition, and pulls images from the file I specify. That's everything I was looking for (and I haven't been able to find all of those things anywhere else).
You can select whether to pull photos from a file or to select only specific photos. That's a nice plus.