My problem turned out to be very simple/stupid: I did not solder the voltage regulator! With that in place, everything works file, so far.

I've found a great way to hand solder those surface mount ICs. If you drag-solder the chips to the board, and then use a power vacuum solder remover (I have a Hakko 808), the joints come out looking great.



I can't manage to get my 0805s to lay perfectly flat, though. Those suckers are a pain!

Good luck!

- Alex