Sounds are not working on Mountain Lion Notification Center
I upgraded my mother's new iMac to Mountain Lion and noticed that the mail sending and receiving sounds were not occurring.  I found the following URL which was a temporary fix, but the problem comes back... so I think there's a bug here.

You might try the following to see if this helps:

http://applehelpwriter.wordpress.com/2012/07/31/no-sound-or-volume-after-upgrading/

Lynn