I’ve got a 1999 Dodge Grand Caravan, 3.8L engine. There’s a sound coming from under the hood that sounds like a drum roll on a piece of hard plastic. No noise on idle or reving the engine while parked. Noise starts upon accelerating, seems to disappear in second gear (automatic trans), reappears in 3rd gear, disappears in 4th. Appearance and disappearance isn’t consistent but seems to disappear when the enging shifts up and the rpms drop. No check engine light.
What you’re describing does sound like pinging or knocking. You could try using higher octane gas next time you fill up.
You might want to check the differential part of the transaxle. There’s a cover.
There have been reports that the pin that retains the spider/side gears in the differential can slip out of it’s hole. The pin then starts hitting the casing of the differential. And if allowed to continue long enough the pin will punch a hole in the differential casing.
Tester
Did try a full tank of premium - no difference.
Then it’s probably not “pinging”, is it? 
I looked around for a TSB regarding Tester’s advice, none found (that does not mean a thing) but I did find a pretty interesting one and it relates to jesmed’s advice. Chrysler says that using a fuel with a “high DI” (drivability index) can cause drivability issues like stalling,stumbling on acceleration,a couple of others (not pinging though) Fuels with a high DI are fuels with a higher octane rating. In short the TSB says, do not use a 91 octane fuel in a vehicle designed for 87 octane. TSB #14-08-97.
How would you feel if you were a Service Advisor that told his customer to “try a higher octane fuel” and he comes back with this TSB?
Tester - On the rid home, tonight, I used “3” on the gear select rather than “D” (I think it limits the transmission to 3 gears rather than 3 and overdrive) and the noise disappeared. I think I’m headed to a local mechanic, this weekend. I’d normally give the repair an attempt on my own but I have no experience digging into the transmission. Any idea on what the fix for this is? (What causes the pin to come loose and how do you fix it - put it back into place or replace with a new one?)
Thanks.
I believe they just ‘went cheap’ and deleted the retainer, like a roll pin or bolt that would normally hold this pin in place. Instead, they used a press fit method. But, due to the forces exerted on this pin, the press fit would allow for the very slow movement of this pin out of position. Resetting the pin into place and the addition of a $3.00 retainer will fix it for good.
Check halfway down this page for some more info:
http://www.turbovan.net/Trans3.html