If you go to a reloading forum and ask this question, or better yet search there instead, you'll see there's a lot of discussion on the topic. Here's what I have gleaned from some of those lengthy discussions.
Vibratory cleaners. Cheapest and easiest to use. Media types most popular, corn cob material and walnut shells. Popular additives, cleaning solutions and automotive wax. Down sides: doesn't clean inside of case well, nor primer pockets if deprimed first. Media becomes contaminated with lead dust. Use with solid lid if indoors to prevent adding dust to atmosphere. Plus sides: Cheap, easy, can also be used to clean loaded rounds of case lube (see below).
Wet tumblers. More complicated, better cleaner. Wet tumbling can be used with or without stainless steel pins. The advantage of using the pins is they will clean the inside of the cases and primer pockets if deprimed to look like new. The disadvantages of using the pins is they have to sorted from the brass afterwards, adding another step to the cleaning process. Also, they are very small and tend to 'escape', ending on the floor and else where. There have also been reports of pins being stuck in priming holes or left in cases and ruining barrels when the case was fired. Note on this last one: I've never experienced and expect negligence on the part of the reloader. Down sides: Much more labor intensive and costly. Can be messy. Brass can actually be too clean, resulting in the need for case lube even in carbide resizing dies. If proper techniques aren't followed, brass can be discolored. If you live in a location with extremely hard water, you may need to use bottled water to avoid deposits. Plus sides: Results looks like new brass. No lead dust or residue as all is rinsed away afterwards. Some add car wash with wax additives and report prolonged storage without discoloration or tarnishing.
No processing. Yes, there are reloaders out there that do not process their brass unless it's gotten muddy. The best I can say about it is it's the cheapest solution (if you can call it a solution). But it does nothing to get rid of lead contamination and leaves cases looking like, well, ****. NOTE: most lead contamination comes from primers, not bullets, unless you are using lead free priming compounds.
Note About Cleaning Loaded Rounds. This is very controversial and if you search the web, you go from extremes like it can blow your gun up to it can't possibly have any effect. The truth most likely lies somewhere in between. The danger does not lie in a round accidentally going off but in changing the burning characteristics of the powder by causing it to break into to smaller pieces. To my knowledge, this has never been conclusively proven, and some reloaders have done pretty exhausting tests, including one who loaded a batch of ammo, set aside some as baseline establishments, then tumbled different lots anywhere from a few hours to several weeks. Firing the lots over an chronograph proved no descernable difference. However, in all fairness, this was one type/brand of powder, and thus may not be representative of all powders. Most will agree that the few minutes (10 at the most, with 5 often adequate) in a vibratory tumbler will remove the case lube with no harm. If you choose to do this however, use clean media, as otherwise you will be coating your reloaded ammo with contaminated dust.
I think that covers everything.
