In the old days we used a dry powder machine that worked like an oversized commercial washer. Great for sheepskins. Immersion in a commercial drycleaning machine would be another way, along with extracting dry solvent on the floor. Everything else uses some water so would be limited moisture cleaning.
Usually... a recommendation of dry cleaning is the manufacturer's way of transferring liability for poor construction to the cleaner. You decide to do a more thorough job of cleaning using water and the rug falls apart or adhesive wicks up, your fault.
Depending on the size and whether or not the rug was colorfast.
Dry foam - treat it like a piece of upholstery that's going to bleed like a stuck pig. Dye Loc, Light Acid- Neutral Shampoo, whip it up into a foam. Apply the foam with a sponge, agitate w/ horse hair brush, extract with a dri-master(may not be practical if rug is HUGE), and speed dry.