2002 B4000 automatic with the V6 4.0L. Some days it doesn't make this noise, other days it does. I've noticed the noise goes away when I rev the engine (or at least I can no longer hear it when the RPMs go up).

 

Sound like the timing chain.

 

How many miles are on your 2002?
Do you do any wrenching (meaning: work on vehicles yourself)?

If so, you can take a peek at the left timing cassette under the left valve cover -it's the one you can check- and maybe see some damage.
Busted up timing components on these engines require removal.

I've been using the "clear flooded engine" method before starting my engine to pump up the tensioners instead of just hitting the starter like normal.
I don't know what varying levels of damage sound like or if your noises are even timing chain related.
MAYBE just your tensioners are worn out.
However, your noises do sound fairly metallic....meaning the big damage is already done.

 

Ok so an update: The noise doesn't happen all the time. I'll start it up one morning and it'll be silent. I'll start it up later that day and the noise will be there. Later that evening, it goes away. Doesn't seem to matter if the engine is warm or cold. If I floor the gas and crank up the RPMs (like if I'm getting on the freeway), there's definitely a noticeable chattering noise. But once I let off the gas and the RPMs go down, the noise seemingly goes away.

There's 116K miles on it. I do some wrenching, but I definitely don't know enough to pull engines and take them apart. Valve cover seems easy though.

 

What's the general condition of the truck? Has it been well oiled?

Ask around for knowledgeable Ford mechanics or, if you're lucky, a busy truck specialist to give it a listen.