The free PowerOff utility had allowed me to schedule Windows XP computers to shut down automatically each evening, but it doesn't work under Windows 7.
Further testing reveals the problem to be that Microsoft doesn't allow services to interact with the desktop directly anymore, so this software is just plain non-functional with Vista and Windows 7. Time to find another free solution...
It's discussed in more detail here:
Windows Developer Center > Windows: General Forums > Windows SDK > "Allow Service to interact with desktop" does not work on Vista
