300E should be VERY smooth and quiet -- shakes can be vac leaks, bad injectors (my next project on teh 300TE), or bad engine mounts (or any combination).
The mounts seem to go every 100,000 miles or so it seems. Having bad ones (that is, empty ones) causes severe vibration because there is a spring in there -- combined mount and vibration damper. Lovely design, does a fantastic job until the glycol leaks out....
Best driveway test is to jack the engine up and check for holes -- the rubber splits next to the steel trapezoid on the bottom and drives up into the body of the mount. In really bad cases (like the TE) the rubber rim at the centerline of the mount will be sitting just above the frame! The 1 - 1.5" clearance is from the bottom of the rubber to the frame, not from the steel to the frame....
1972 220D ?? miles
1988 300E 200,012
1987 300D Turbo killed 9/25/07, 275,000 miles
1985 Volvo 740 GLE Turobodiesel 218,000
1972 280 SE 4.5 165, 000 - It runs!