Finally got the sound working on my laptop, running Mint 12 and Cinnamon.
The fix was to go to the Gnome Alsa Mixer and uncheck “External amplifier”. The “Sound” app on the system bar seemed to allow the same change, but it didn’t seem to stick.
I first tried adding more gStreamer plugins. No effect. There were already some gStream plugins installed, but there are lots and lots of others to distract you!