Is there a simple way to undo an app update? I'd like to revert to the previous version and stick with that permanently. Is that possible?  

I have Beautiful Widgets installed, and had it set to auto-update. It just did, and the 'update' is totally unacceptable.  

I've got several devices, all rooted and running different versions of Android. Some have auto-updated the APK to the new version, some have not (I just disabled auto-update on those that have not).
Android does not natively back up applications so you can't "undo" an app update. Best thing I'd say you can do is something like this howto. It backs up your apks and their settings. It uses Titanium Backup and requires root, but rooting is not your problem.