I lubed the chain and all the pivot points and rode over to get pizza a couple nights ago and I didn't hear the noise anymore. Then, this weekend, I was going on a longer ride and I heard it again, louder. I stopped and looked down and saw that my freewheel was flopping around.
Apparently, the lockring on it loosened itself up enough to let all the bearings fall out and then loosened the rest of the way so the lockring was just sliding around on the axle while the freewheel slid back and forth. I took the back wheel off and was able to get the lockring back on tight enough by hand to hold it together while I limped back home.
The freewheel is (was) a Suntour Pro-Compe nickel plated deal. I've read more about it since then and it sounds like they had a tendency to unscrew their lockrings by themselves and have this happen. Just thought I'd post an update here in case anyone has a similar problem in the future.
From Sheldon:
http://www.sheldonbrown.com/freewheels.html
Servicing Freewheel Bodies
Some SunTour Pro Compe and Perfect freewheels tend to unscrew the lockring. Fortunately, only the bearing balls will fall out: the axle and dropout will retain the other parts. It is usually possible to screw the lockring back in (counterclockwise) and ride home as long as you don't coast.