I do think some more specifics about the noise might be helpful...I have to tell you though, your description does make it sound an awful lot like a loose catalytic converter heatshield...whether it happens when you decellerate or let off the clutch. I had this happen on my old
95 Civic, my sister's 94 Civic, two of my friend's cars, and I know it's a common problem in Miata's as well. My mechanic says it's a problem with all Japanese cars but that it is quite alright to just rip it off and toss it away. Is it a buzzing noise? A loud RAT TAT TAT? Get graphic and mayhaps we can give you a good starting place.
Two good sounding methods for under the car:
Start the car, lie on your stomach and look under the car. Have someone (sounds like maybe your dad?) sit in the driver's seat and...in neutral!...slowly push the throttle in. If, as you say, there's a rattle as you're letting off things--clutch or throttle, you're not clear--then it's likely that part will vibrate at a certain frequency, which you can replicate with the revving of the engine. Just make sure the driver doesn't redline under no load! If this method works, it probably won't take much throttle.
Another method I came across over on the ineffable
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is to jack the car up and start tapping things (exhaust, chassis, transmission, clutch assembly, etc.) with a rubber mallet. Eventually, you can track down what system is making the noise, and then where in the system the noise is.
Also, do you have any aftermarket (or OE) bracing that might be coming loose? Now's a good chance to torque those bolts...and heck, change the oil filter while you're down there...
Best of luck, keep us posted.
Cheers,
-r0ll