Unplug everything you DON'T need to POST, such as sound, HDD, DVD/CD, floppy (lol), and all but one stick of RAM. Of course, make sure to check your motherboard manual to see where that remaining one stick of RAM should go. Double-check your plugins to the motherboard. Then try to fire it up. If it still doesn't turn on, you could be looking at anything form a PSU to motherboard, to that one stick of RAM, so try another stick in the same place. Then try another slot. If still nothing, try a different PSU. A PSU can be dead even if an LED lights up. There is more than one power line to the board. If one fails, not all fail. Know what I mean? Ultimately, use the process of elimination to figure out what is wrong. Eventually you'll find out.
I have this happen to me occasionally and, although it's annoying, my problem has a simple solution. Apparently mine is not a permanent hardware issue, but a temporary confusion in the BIOS. It usually starts because something caused the system to freeze up during normal operation and I have to turn off to reboot. But the damned thing won't respond in any way to the front button! However, if I pull the power supply cord out or even just reach around the back and turn off the manual switch on the back of the PSU, and then leave it off for 2-5 minutes before restoring power, it all works again! Push the front button and it boots up normally right away. Whew!
My hypothesis is that somehow the BIOS ends up in a confused loop. When the machine if "off", there is still a small part of the BIOS running and one of its functions is to monitor the front on / off button to turn on. If that monitoring process somehow freezes the machine won't respond to a button push. But if you really do power down the system for long enough to lose any output from the PSU's capacitors, the BIOS loop will be shut down. Then when power is available again the BIOS will re-initialize itself normally. Give this a try on your machine before opening up and moving components around.