thrust arm bushings are a common ailment. The E34 is really fussy to front end parts as in if there is ANY play anywhere you will have all sorts of vibration problems.

Does the shake get worse when you apply the brakes? If so I'd shoot for replacing the thrust arms first, post a thourough front end inspection for work parts.

Look up "nut behind the wheel" as that is a common source for play.

When I had serious problems the tires were ****ing square, they would visably move up and down as spun on the balancer, which the tire place said didnt matter, 2nd set of tires were the same. Winter coming I got some 60 buck each semperit snows made in Germany...smoothest tire ever owned....tire place sucked. Now im running the cheap falken tires and its still pretty smooth.

I never go to the same places anymore.