I just upgraded two machines to Mountain Lion. The first being a Mac Mini mid 2011 with no problems. The second being an aluminium Macbook. The Macbook audio was dead once installed, all I got was a cross out audio option when trying to increase or decrease the volume. In System Preferences the only thing that showed was headphones and even they didn't work.
I am sure Apple are working on the fix but for now you can do this:
Open Activity Monitor found in Applications -> Utilities folder.
Click on the CPU button at the bottom of the Activity Monitor window. The one to the left of System Memory.
At the top change the drop down that says 'My Processes' to 'All Processes'
In the search box next to that type this in, or part of it at least. Even typing 'core' will find it.
Search for coreaudiod
Select it and then click 'Quit Process'
You might find it doesn't quit. I clicked quit and force quit. Either way once you have done this you should find your audio is back.