Booming Cressida

We have owned a 2001 Toyota Cressida Grande for a number of years and apart from of what I describe as a "booming" noise under acceleration we have been delighted with it. The car is in perfect condition despite its 200,000 km, but we haven't been able to fix the problem despite a number of attempts by a number of mechanics. Twelve months ago I acquired a 2001 Cressida GLX in concours condition, which is identical to the Grande mechanically, but doesn't have the noise irritation. Can you or any readers suggest a solution?