I have two MDD's exhibiting the same condition - the power light activates when pressed, but goes dim when released. There's also no chime and no fan activity. The state of the PRAM battery doesn't matter, whether it's used, depleted, new, or even absent. The state of the four RAM slots doesn't matter, whether they're empty, partially full, full, or contain known-bad memory.
I've read that there's a capacitor inside that may be offending. It's 390 uF and 450V at 105 deg. C. It's also 35 mm in diameter and 42 mm in height. I forgot the manufacturer, but recall the name amongst a known set of manufacturers of defective capacitors. Even though there are no visible signs of wear, this person replaced the capacitor and was able to fix his G4.
I'm going to badcaps.net and hope other people can support what this other person experienced. I believed I pull the original story from the Apple Discussions Forum.