I, too, am having this problem. I'm running a 2010 Macbook Pro (15") and just to be sure, did a complete backup of my stuff, reformatted the computer, installed Mountain Lion fresh, and manually replaced my apps (reinstalling them and copying over their preference files) and personal data. I have a system lacking any corrupt preferences, cruft, or anything else that I can tell might impact this. I've run OnyX to deep clean all the caches and repaired permissions. The disk is reported as healthy with no problems by Disk Utility.
Notification Center will (99% of the time) hang if I ever try to open it after the computer sleeps. In addition to notification center hanging, it will also cause the Dock process to hang as well. My mouse will be unable to interact with any windows (no clicking or ablity to scroll). Thank goodness for Alfred, wherein I can quickly launch terminal commands to restart things. As an aside, is there no keyboard shortcut to kill a process in Activity Monitor? Opening that, I can use the keyboard to shuffle through items, but since I can't click on anything, I can't seem to kill any processes.
If I manually kill Notification Center, I can finally interact with windows again, clicking, scrolling and typing all work fine. Unfortunately, I still need to kill the Dock process. When I run sudo killall Dock, instead of responding instantly, as it normally does, it can take up to 5 minutes before the dock process restarts, so I'm thinking there's something fishy going on betweeen NC and the Dock when invoked after sleeping. I'm not an expert in this stuff, though, so maybe Apple can take note of this.
Again, this is all happening on a clean install of 10.8.2. I seemed to notice the problem [i]after[/i] I linked my Twitter/Facebook accounts to OS X, but can't confirm that. Hiding the share icons and unlinking my accounts seem to make no difference. It would be interesting to know if all the people here have linked social networks and share buttons or not.