You can drop them all into a water tight container like an old GatorAid bottle, add some hot water and soap. Screw the cap on tight, then give them a good shaking, then a good rinse. Let them dry and presto. You can stand them on a hotplate and cook the water out of them, or stand them in an old skillet and turn on heat until they stop bubbling.... or set them out in the sun, or just let them dry. They will dry eventually one way or the other.
I think the buffalo hunters would boil their cartridges for a while to get the funk out of them, then dry them by the fire.
You can polish them with some fine steel wool (#0000) if you want them shiny. You don't have to have a tumbler, it just makes it more convenient. As a matter of fact, you don't really have to clean the cases, but it doesn't hurt to clean them to get rid of old powder residue and what ever else that may get into them.