I recently just moved my desktop back to my apartment after a few days at my parent's house, and it no longer turns on. When the power cord is connected, the fans occasionally spin and LEDS will still flash, but the computer just doesn't turn on. My friend who is good with computers tried helping me out with the problem, and the first thing he tried was unplugging everything but the motherboard, RAM, and the cpu, which didn't work. Next, he unplugged the cpu power - and the computer turned on, but didn't really go anywhere (no POST, I think is what he said?). He suspects that there might be an issue with the cpu-mobo connection or that the cpu is bad- would this cause the computer to not turn on if the cpu power was plugged in? Might it also be an issue with the PSU?
Whenever there's a situation like this, the PSU is the primary suspect. However, it could be lots of things.
Its rather odd that this happened after being moved. Makes me think you jostled something loose.
So, remove and re-seat everything . . . memory, expansion cards, both motherboard power cables. Check that the cpu fan/heatsink didn't come loose . . . grab it and (gently) try to rock it. Check that the cpu fan plug is firmly in its socket. Check that the front panel Power switch cable is snug on its pins on the motherboard.
If you have a reset switch, REMOVE its connector from the mobo if you can.
(EG) A Dell will not power up if the cpu fan doesn't spin. So, power up and immediately try to help the cpu fan spin. Sometimes a fan will develop a bad spot that prevents it from starting to spin, but it will spin if you get past that spot.
If none of that works, swap in a known working psu.