Well, we might be insane, but we're not crazy.
If you've converted to a battle.net account, you cannot download the patch via the traditional log-in-and-it'll-start method. To get the patch to download, enter your old non-battle.net username and password and the game will begin downloading patch 3.1.1. It's about 10-megs.
You can't log-in and play the game with that user name and password anymore, but you can get the patch with it.
Seems that Blizzard hasn't worked all the kinks out of the battle.net account system. But at least there's a way around the problems this time.
Update: Articles goes up...and 45 minutes later it's fixed!
Many thanks to our twitter followers who let us know about this!