My 94 Miata just started making noise. Different noise then the lifters (deeper then the clicking noise). When I put a slight side pressure on the gear stick the noise stops.Only happens wehn in gear and accelerating. Stops as soon as I let up on the gas or put it in neutral.May be unrealtated , but the Miata shopchanged the tornboot under the center console while it was in for other work but the nisedidn't start fora couple of months later.

 

I would have to guess a bad engine or tranny mount... especially the tranny mount in your case. It is a "94 after all and if they haven"t been checked or replaced by now they are no-doubt due.