Car is shaking when accelerating?

This just started happening and I don't know why. When I accelerate the entire car vibrates and it ceases when I'm not accelerating. At constant speeds on smooth road it doesn't vibrate, but going up hills it does, so this tells me it is only when some torque is being applied. Engine is running great and I have brand new tires on that were balanced. My car is a 96 Dodge Intrepid 3.3L, automatic transmission.