Sometimes I could unplug the power cord, drain out power, do a voodoo dance, and similar things like that, to get it working again.
After about four returned monitors, the problem seemed to disappear. (Well, I was being extra careful unplugging the monitors.)
Anyway, the monitors are back to their old tricks. This time, I found a nice CD ISO that you run to reset the monitors.
It's attached at the bottom of this post. Enjoy
*Make sure the 'broken' monitor is plugged in only by DVI, and boot off the CD provided.