mpd: Failed to read mixer for 'My ALSA Device': no such mixer control: PCM

@CL was right; alsa was working but mpd was not.

I discovered that if I switched to the home folder of the mpd user (export HOME=/var/log/mpd), the aplay commands (e.g. sudo -u mpd aplay /usr/share/sounds/alsa/Front_Center.wav) would no longer work.

More Googling until I found http://ubuntuforums.org/showthread.php?t=1138454

The solution was sudo apt-get remove pulseaudio. Once that was done, everything was working again.