Strange power-up problem

I have a GA-7VT880-L motherboard. (Socket A CPU with VIA KT880
chipset). Recently it has developed a strange problem when powering
up:

Press the power button, and you can hear the power-up sequence
starting (CPU & PSU fans start, disks begin to spin up). After about 2
seconds, it just shuts down again. After that, pressing the power
button does nothing at all - the system is completely dead. However,
if I then switch off the PSU (it has an on/off switch), wait a while,
then switch it back on, the system will usually power up successfully
and can then be used fine. (Sometimes I may need to do the PSU on/off
cycle more than once).

I replaced the power supply and the problem persists, so I'm guessing
it's some kind of motherboard issue. The question is, does this sound
familiar to anyone and if so is there anything I can do about it? Or
do I just live with it until the time comes when it bothers me enough
to replace the mobo?