So, to test a theory on how to fix graphical tearing on a Xubuntu 12.04.1 install, I tried installing compiz.
Not only did it not help, but it also broke the desktop environment. So I uninstalled (followed the reverse of the compiz installation steps I did). Uninstalling though doesn't clean up everything, and the desktop was still broken.
Fortunately, I found this post which suggested removing ~/.config and ~/.cache.
You don't need to zap those entire directories though. Inside each one there is a subdirectory that starts with "compiz". Those are the directories that need deleting. After that, I rebooted and the regular xfce4 desktop was working again.