Dutch's back flush method works very well. The sudden rush of soapy gray tank water loosens up the stubborn material. However, give the gray tank a few seconds to flush out before using it. Heavy solids (like coffee grounds) accumulate at the bottom of the tank and heavy solids are what you're trying to get rid of.
You don't need the extra flush king valve to accomplish this. Simply hold the stinky slinky up higher than the level in the tanks and open both valves. Let the gray water run into the black water for a few seconds (if you have a clear section you can see when it stops), close the gray valve and drop the hose.