I isolated the problem to a rattling in the back of the freezer. I tried tightening the screws and put a lightweight insulation strip between the various plastic interior parts. That worked for a bit, but started making noise again in a few weeks.
Ultimately, I identified that the evaporator fan was the source of the rattling; the fan was worn out, and caused a lot of vibration when it was running.
The part was very straightforward to replace. Consider taking a picture before you start, so you can connect the wires in the right place.
My only hitch, and this wasn't the fault of ApplicancePartsPros (their customer service was great, BTW), was a bent fan bracket. Both the original part, and the second one they sent out, had a problem. It turned out to be easy to fix; simply disassemble the part, and straighten the bracket with a pair of pliers. No big deal, and the new part is very quiet. My wife couldn't believe the change.
Good luck!