I have Game Alert Preferences set to "play a sound when it is my move", but no sound plays.  The Meebo chat sound works just fine, and I know it's not a hardware issue.  Any thoughts?

Also, is there any way to customize that sound, if and when I get it to work?

No ... and yes.

It should be possible for everyone implement this locally ON THEIR OWN computer. The chess.com window title changes from:

1. My Online Chess - Chess.com (assuming you've clicked on PLAY -> Online Chess) to
2. xyz vs artfizz - Online Chess - Chess.com - when you're viewing a game, but it's not your move. Then to
3. MY MOVE! - xyz vz artfizz - Chess.com - when it's your move on that game.

Using a scripting tool (e.g. AutoIt) that can detect window titles, run a .WAV file (usingSound Recorder) when a browser window (or browser tab) is detected that begins with "MY MOVE!".