I won't even pretend to understand this, but I thought I would share it in case anyone else runs into this problem.

My system:
Gigabyte GA-EP45T-UD3LR motherboard
Ubuntu 13.04 64-bit clean install
Adobe Flash Player Plugin Installer installed
Sound output via Toslink (digital optical) cable

After weeks of working, today I noticed that although sound effects and music (via Clementine) were working fine, I had no audio in Flash in Firefox.

I reinstalled the Adobe Flash Player Plugin Installer, but still no sound in Flash videos.

I looked at my sound settings, and found that on the Output tab, "play sound through" had the digital option selected, but the digital option was listed as "Digital Output (HDMI)" instead of "Digital Output (S/PDIF)" as it used to be.

I clicked on the "Analog Output" option, and when I did, the digital output option changed to "Digital Output (S/PDIF)." When I re-selected the digital option, I had sound in Flash again.

Very weird, but it's working again now. Hopefully, this will save someone else some trouble.