Although I see an option in System Preferences > Notifications to enable applications to pop a sound on new notifications, Mountain Lion is not actually playing those sounds.
Has Anyone run into similar problem?
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:
Sorry my terrible einglish. I have same problem exactly. The sounds of some notifications of some programs of the center of notifications do not work (Capucin, Tweetbot), however others works o.k. (mail,mensajes). I think that many applications even aren'y ready to work with the center of notifications especially the sound the sound.