I am also having this problem with my fresh Mint 13 (64-bit) Cinnamon installation. I am hearing the 'welcome' sound which plays when booting into Linux, but can't hear audio from YouTube, MP3s, etc.
Funny thing is that the sound was working fine when I installed Mint 2 days ago. However, yesterday I had to reformat my HDD and reinstall Mint 13. Some setting/package has obviously changed between the two installations. Perhaps it is one of the MintUpdates I downloaded?
edit: I'm not sure what happened, but my sound is now working properly. I'm not sure what did it, as the most significant thing I have done over the past 5 hours is installing VirtualBox (with Windows XP image), and a handful of Linux Mint software including Exaile. Consider this problem solved for me, and let's try to help the OP.