OK, thanks to my friend Bill who helped me understand how Akai wired this thing up, I fixed my 'noise in the headphone jack' issue.
I'm attaching a couple pictures to hopefully make it clearer (sorry for the slight blur in the 2nd one).
Anyway - the summary is I cut the headphone jack ground away from frame ground and jumpered it to analog ground. That gave me no background whine, no tape sync noise in the headphones, no thumping in the main L/R. It also gave me a very clear, clean, and loud headphone signal into my AKG 240DF headphones.
If you want to do this yourself, I make no claims as to whether your problem is exactly the same as mine nor whether these same steps will actually fix your MPC. I also warn you not to stick your hands inside an MPC while it is plugged in.
with that out of the way...
Open your MPC, unplug the 5 cables connecting the top output board to the rest of your MPC (don't forget that thin 2 pin connector).
Unscrew the 2 screws inside holding the output board to the indivudual 8 outs board.
Remove the 3 screws holding the output board to the back panel (don't forget the screw above the digital jack).
Flip the board upside down and locate the headphone jack and the ground pin of the headphone jack. Notice that the ground pin is soldered in the middle of a small square that is connected to the larger silver section by 3 tiny bridges.
Using an Exacto knife, a box cutter, your Aunt's sharpest paring knife, old rusty tweezers, or anything else you can think of, carefully cut those small metal bridges.
Yes, I know you were careful, pressed hard, and scraped away for a while, but I still don't trust you. Use a multi-meter to verify you have no continuity between the ground post and the surrounding area. It's very easy to push those metal bridges around and not actually get them completely out.
Now take a small piece of wire (even use part of a resistor leg or capacitor leg) and carefully solder it to the ground pin of the headphone jack and then to any analog ground point on the board. Be careful that solder doesn't reconnect the areas you just tried to separate. Recheck with your meter!! I happened to choose the analog ground side of capacitor C200, but you can choose any suitable point.
Check your work, reassemble, and test.
Let me know if it works for you.