***Update: There is a workaround to play Diablo III if you don't mind:
1. Not playing with your friends 2. The possible lag. 3. Creating a level 1 character
USA players can play Diablo III by going to the login screen, press the Options button. Select Account. Select Europe from the Server dropdown menu. Happy hunting. The only setback is your character list won't appear in the Europe server. Try the USA servers every time you leave a game. The developers are currently investigating the Error 3006 issue, and implementing a fix at a later time. No ETA.***
Blizzard acknowledged a known issue with Error 3006 time outs. I experienced this a few minutes ago. Powercycling didn't help me, but you might have better luck.
[bluepost]This is a known issue, and our development team is working on a resolution. Customer Support cannot assist with this error, although we do recommend powercycling your router and modem as detailed on Step 2 of our Diablo III Connection Troubleshooting page.
As this error can be caused by temporary server issues in a specific region (Americas, Europe, or Asia), you may be able to switch to another region to play while we work on the servers in your original region.[/bluepost]