My Flash videos were silent. Audio seemed to be working otherwise. I reinstalled Flash, but silence still reigned.
Mac OS X Hints had several old fixes: A possible fix for no sound in Flash on Intel Macs. The most direct fix involved running a Midi utility to correct the sound sampling rate, but my sampling rate was fine.
Turns out there's a wee little sound control in Flash video. It's independent of the system sound. Mine had somehow been set to the lowest value.