We just bought a 2001 Dodge Caravan and had the same problem. We went back to the dealership and one of the young guys there discovered it was caused by the luggage carrier bars on top of the van. The two cross bars, when removed, will get rid of the whistling noise. .
It sounds like it may be an exhaust leak, upon aceeleration, more exhaust escapes and makes a louder noice. If out of warrenty, take it to a muffler shop that does custom exhaust work, because a crossover pipe is a crazy cost from a dealer.
To improve a bit on the first answer. The cross bars should have rubber inserts to close the gap in them. If these are missing you'll get the whistle. Yes removing the bars will eliminate the sound.
My 2003 has what I think is your issue, it is alternator whine amplified through the stereo, even when the radio is off you here it. The only time it goes away is when you actually remove the radio. This is caused by a bad alternator. You can help remove the noise with an electrical filter (capacitor to ground).
Air conditioning belt: I had an annoying screeching sound on acceleration which turned out to be the a/C compressor belt which I replaced for $10. It started making the noise again so I tightened the belt and it went away. You can test to see if this is your problem by turining the A/C or defrost off when the sound comes.
In our case (2003 model), we replaced the fuel pump which was located in the fuel tank. Easy changeout.
I have noticed this on my '03 Grand Caravan, and have found it to be the fuel pump. I really notice it when it is hot outside. The dealer said they wouldn't touch it until it goes out. They have replaced them and the new ones will start to do this sometimes. If the dealership told me to wait, I will. They are usually known to try and take all the money they can from you.
Could be in need of a tune up. Replace spark plugs, wires, air and fuel filters.
check your swaybar links and bushings
Loose belt - will sound like a screeching noise.
The noise can be comming from the weather stripping around the door, the wind passing past the antenna or from the wind around the luggage rack on acceleration.
Have the shop tha installed the motor mounts check and see if all mounts were replaced and if they are tight.
Check or replace axles, worn axles are primary source of shaking when accelerating.
No answer for you - just want to state that our 2002 Caravan recently (7/09) starting making a load clicking noise when it is turned on. Belt looks good. I'm guessing its the compressor or clutch.
Do you have a check engine light on? I would consider the idle air control motor to be a likely culprit. You also might listen for the whistling sound of a vacuum system leak.
This tranny has a the feature of putting itself into "limp" mode when it senses a problem with itself. Limp mode is full-time second gear.
Some adjective to describe the noun caravan are: a desertcaravan, a traveling caravan, a Bedouin caravan.