What autopilot do you have. -50 or -150?
The older autopilot has (had) a rate gyro in the Pitch Channel Computer that'll cause exactly that problem. The newer uses the attitude input from the vertical gyro or whatever attitude system is installed.
All autopilots require elevator position input from position transmitters in the top of the tail. They must be synchronized to within 50mvac of each other, have a less than 50mvac (the closer to zero the better)null at neutral and their respective nulls must be on the same side of zero. That is the most common cause of a short cycle porpoise or pitch oscilation. Ask FEDEX about their experience in the early 90's when one of their engineers decided that the fleet needed to have the elevator position nulls reset and didn't follow the three rules.
The air data computer will also cause pitch oscillations.
I once had an aircraft with an ADI loading down the circuit and causing elevator flutter.
The vertical speed wheel in the control panel is a possible but unlikely cause.
Since you seem to have a problem with turbulence you have a problem with something that reacts to pitch attitude changes and the response isn't appropriate to the input.
Two important rules a lot of people forget: 1. Flutter or oscillation is caused by position input (the system doesn't know where the elevator or the aircraft attitude is) and 2. Hardover or excessive movement is caused by a rate input. (the system doesn't know how fast the elevator is moving)
The very lasts thing I'd look at is the power control cylinder.