Can anyone give me some pointers as to where the solution to my problem may lie -

My car occasionally starts shaking under my very normal commuter-style driving. The shake starts gently and then builds up so that the whole car is vibrating (apparently from the rear though I can't be certain and it certainly can be felt through the wheel though it is not like a wheel-wobble shake, rather the whole floorpan feels like it is shaking underneath the seats) and at 80ks an hour on the motorway it is very uncomfortable with the car feeling like it is going to shake itself to bits. Then the shake disappears; either gradually in the opposite way to how it starts or, if the engine is stopped (e.g. for refuelling or to phone my friendly neighbourhood mechanic to ask him about the problem) and then restarted, the shake is gone!

The source appears to be drivetrain because if you put the engine in neutral (it is an auto) so that the revs drop right down the shake is unaffected.

My mechanic said it might be a tyre - but they are all in very good condition and I don't understand how shaking caused by either tyre or wheel balance (his other suggestion) would come and go in the way it does.

My car-savvy brother-in-law said it wasn't likely to be tyre-related.

The shaking feels as though you are driving over a rough surface - e.g. over a surface like a continual rumble strip.

Any suggestions greatly appreciated.