I managed to scrounge up yet another dead LCD monitor today. This one sat on a desk for about three years and had a relatively uneventful life, yet managed to fail anyway - with the exact same problem as the ViewSonic one I got a few weeks ago, no less. A real testament to how they don't make them like they used to!
(For clarifications, though I thought I had fixed the ViewSonic monitor, that actually turned out to be a side-effect of heating it up with the hairdryer. Once it cooled back down, it went right back to shutting off.)