It seems that this is a fairly common problem, one that my 1999 Ford F-250 Superduty Supercab V10 that developed as well.
It took some head scratching and trouble shooting, so I figured I would post here in case it helps anyone with a possible solution!

Symptoms Included:
- A clunking / thud noise when coming to a stop, taking off from a stop, and going over bumps.
- Noises were especially noticeable when pulling in or backing out of the driveway.
- A feeling as if the weight of the truck were shifting forward or backward over the front axle.
- Vibration/Feeling in the driver side floor and pedals, particularly the brake pedal.

Not the Problem:
- I had already installed new Moog greaseable upper and lower ball joints a few thousand miles earlier, and inspection showed no play.
- My mechanic had just installed a rough country front leveling kit (all new leaf springs and bushings), new track bar, new Bilstein 500 shocks, new brake pads and rotors.
- The body mount bushings looked surprisingly good for a 1999 and we did not suspect them to be the problem.
- We took apart the drive shaft, checked everything and put it back together. Nothing out of the ordinary. The problem was the same in 4x2 and 4x4 modes.

Problem #1:
Close inspection of the driver side front brake caliper showed the bridge to be worn out. Even with the bolts tight as could be, there was a fair amount of play. I could take my hand and rock the caliper side to side and front to back. We installed a new caliper assembly and this made a noticeable difference, eliminating a bit of noise and vibration, especially when braking, but the overall issues persisted.

Problem #2:
The sway bar end links showed a fair amount of play. We replaced them with new Moog brand parts, and there was a noticeable improvement.

Problem #3:
The original factory sway bar bushings were worn and had a large amount of play in them. Had the mechanic install new aftermarket sway bar bushings (don't remember the brand but they were blue), and not only did this not help, but seemed to make the problems worse. Drove around for a few days thinking they needed to be broken in, but the problems only got worse. Took it back to the shop and while the truck was on the hoist, the mechanic put his hands on the bumper and pulled down onto the truck a few times. Lots of noise came from the new bushings! We swapped them out for Ford/Motorcraft bushings, and upon the first test drive, all the problems disappeared. I was reluctant to believe that the sway bar bushings could be the source of such significant problems, but they were. Spend the $ and go for the Motorcraft parts! We went with a cheap set of aftermarket to avoid the wait from the local Ford dealer who had none in stock. Poor decision!

Other Noises:
Discovered 2 other sources of noise during the elimination process. My tailgate hinges were squeaking, some lubrication took care of this. My spare tire (factory mounted under bed) had become loose and was wobbling, sending noise around the underbody of the truck. Both of these are worth checking on!

My truck still has other noises (rear doors of super cab, rear seat, etc.) that I need to tackle, but it's a heck of a lot quieter now and rides great!

 

Had this problem a few years ago on my 2001 F-250 4x4. Ended up being the drag link (large rod tying the front knuckles together). Both ends worn out and I could feel it under my feet when crossing railroad tracks, potholes, basically any size hole I hit.

 

When you think about how much pressure is being put on the sway bar and the bushings, it makes sense they could be part of the problem. On a work truck they were so worn you could see light through them and easily flop the bar up and down when disconnected from the end links. A high quality poly bushing kit is the way to go.