Hey Sparky! Another Antec
When the PSU is unplugged for awhile it will reset itself, then it will power-up the system.
Get rid of that 350 SmartPower now! It just might kill a HDD on it's way out.
If it's less than 3 years old, RMA it and have a spare PSU, for whatever but not for a PC.
If it's over 3 years old, open it up and find leaking/bulging Fuhjyyu capacitors
eWiz.com has some Enhance 360W units for less than $30 and the ENP-5140GH for less than $60.