I also have a cleaner and use the water, vinegar, dish soap solution. It works just okay, nothing spectacular, but they are clean around the primer pocket and inside.

I usually then run them through the vibratory and polish them up with the media and Nu-Finish, then they are SHINY!!!

I have heard recently though that stainless steel 'pins' used in a tumbler actually does both, cleans pockets and inside and polishes to a bright shine. Some of the pictures that I've seen attest to that too. You still need to use a soapy water solution, so there will be drying involved after tumbling.

The best part is that there is not much in the way of dust if these methods are used.

Stainless Steel media procedure with pictures - Calguns.net