I've got the same problem (early 2008 Mac Pro 8 core 2.8ghz). It boots up if the power cable has been removed for sometime, and runs fine. The diagnostic led's on the logic board do not indicate any problem, but for a few weeks now I have found the computer off after leaving it sleeping over night. This seemed to happen more often, and it would take longer for the chimes to sound after turning it on. I've ordered a psu, so fingers crossed :-)
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== Update ==
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I fitted a replacement power supply a few days and it has fixed the problem. So I would say that if you have the same issues as Hitcher, it's the power supply.
I've got the same problem (early 2008 Mac Pro 8 core 2.8ghz). It boots up if the power cable has been removed for sometime, and runs fine. The diagnostic led's on the logic board do not indicate any problem, but for a few weeks now I have found the computer off after leaving it sleeping over night. This seemed to happen more often, and it would take longer for the chimes to sound after turning it on. I've ordered a psu, so fingers crossed :-)