|This isn't ours, but if you have an access pipe on your a/c unit somewhere, pouring a 50/50 water/bleach solution down it every few months or so could help prevent major problems.|
The bad news was that was the extent of my diagnostic ability. Fortunately, I got a recommendation for a good company from a friend, they were able to come out this afternoon, and the air conditioner is now working again. (Just in case you were wondering, it was a blown transformer caused by a short in the wire out by the condenser. My guess is the damaged wire was caused by a lawnmower or weed whacker that got a bit too close at some point in the house's history.)
Here's hoping that none of you have repairing an air conditioner on your To-Do List!