Mine would do the same as the OP. It was the Transmission mounts. Funny enough, I just recently heard an exhaust rattle as well, looked around under there, and found one of my hangars had come out. The rubber bushing inside the rear hangar mount was bad. My local muffler shop had a bushing that worked and $10 later, they had it installed and hung properly.
Welcome to the wonderful world of YJs.. you'll soon learn how to listen, feel, and smell your way into vehicle diagnostics.
♔ 91 YJ "Alice" 4.0L, AX-15, NP 231 Flange, Adams Driveshafts F/R, 2.5" OME 36Rs, 5/8" /f12 Booms, 33" Yoko ATs, 4.10 D30/35 Aussie Lockers F/R
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♘ 05 LJ Rubicon "Ghost"
♠ Luck Favours The Prepared ♠