My achieva made the same noise and it was the transmission mount was past do for replacement, also it could be your lower torque strut mount on the bottom passenger side of the motor. I had to replace both of them and my noise went away. Also the torque strut mount it pretty easy to replace if you know what you are doing, but that mount is very small and only can be accessed by pulling the passenger side tire and the removing the plastic shielding that covers up the harmonic balancer and a few other things. The transmission it one that will take you awhile to do because it is a two person job to remove the bracket that the transmission mount attaches to. But for me the noise you were describing was the mounts but it could be something different on your car. I thought I would just give my helpful advice.
1996 Oldsmobile Achieva SL
White 4 Door