Just in case this saves anyone else a few minutes of boring googling: the solution was to log out of the iTunes store and log back in again.
iTunes on Mountain Lion was being very unresponsive and beachballing, despite not apparently doing anything except playing music, and Activity Monitor said it was using 100% of one of the CPUs.
The CPU activity would drop right down when I stopped the music and jump up again when I restarted it. After searching around on the Apple support forums, I saw the suggestion to log out of my iTunes Store account; it seemed unlikely to work, since I didn’t have the store open… but it did. And I was able to log in again without the problem returning (yet! *crosses fingers*). Hope this helps someone.