won't boot-up unless hold power button for several seconds

I have a Dimension 4700.  I purchased it 6 years ago, but it has a new harddrive, video card, and memory, and it's running Windows 7 Professional, 32 bit.  Otherwise, it has all the original hardware. 

I recently started having a problem while turning on the computer--whether from off, stand-by, or hibernate.  When I press the power button, the screen stays blank and nothing happens in the computer except for a spinning sound--either from the fan or the hard drive.  The only way to make the latter to turn off is to hold the power button for five or more seconds.  The computer will do this until coming on with the third or fourth attempt.  I have discovered that by holding the power button down for two or three secons at first, though, that it will boot up normally.

I have run Western Digital's diagnostics on the hard drive and Dell's Diagnostic Utility on the hard drive, memory, chipset, etc.  Those diagnostic checks have found no problems.  Before going too far in replacing other components like the power supply, I'd like to see if anyone has ideas on what the issue could be.