It turns out my D200 autofocus problem was something simple and I was able to fix it myself.
The autofocus mechanism on an SLR camera is driven by a small servo motor in the camera body. This motor turns what looks like a small flat-head screwdriver blade, which in turn turns a small screw set in the edge of the attached lens, and this turns the focus of the lens.
The problem with my camera was the driver was stuck down for some reason. It is normally spring loaded and is sticking out a bit, just enough to catch the screw in the lens when the lens is attached.
Here's a picture that illustrates it better than I can describe:
Anyhow, rather than sticking out like it is in this picture, the blade was recessed into the hole. So, I just took a very thin pointed tool and poked at the blade. When I pressed on it, it popped back out. I'm assuming when I originally turned on manual focus, the blade was retracted but then got stuck in that position for some reason.
Hopefully the problem won't occur again, but at least I now know it's pretty easy to correct.