what is this problem
Might be your Transmisson mount. That kind of describes the clunking I heard when my transmission mount was torn in half. Couldnt' tell it was torn in half until I pulled it out of the car.
Obviously I'd put the car in the air and check the Shocks, Springs, LCA's sway bars, torque arm and end links and make sure everything is bolted up properly.
Also while driving try to see if you can determine more precisely where the noise is coming from.
If you can repeat it at low speeds, see if someone standing outside the car can tell what side it's coming from. or if it seems to be centered under the car.

