Pulled the left front wheel tonight, and the source of the ticking was apparent immediately. There are two pins that hold a metal spring plate, which I assume hold the pads in place. One of the pins had shed the wire that holds it in place, and had worked it's way out of the caliper, rubbing against the back of the wheel. Glad it was a easy (cheap) fix.
Has anybody else ever had this happen? The wire that goes in the end of the pin now seems pretty snug, but I'm afraid it's going to work loose again, so I will be replacing both sides next week.