Keep in mind that primers have mercury in em...
Uh-oh, I think you just confirmed your age!
Experience with fulminate of mercury based primers generally means you qualify for senior discounts.
Mercuric primers haven't been used in decades. Besides the toxicity, the mercury weakens brass, making it dangerous to reload. Primers are fueled by lead based compounds these days, except for some factory "lead free" loads for indoor shooting.
Back to the question, I use dish soap, hot water, and a brush to scrub my black powder cartridge cases. For smokeless loads, I would just wipe the worst of the dust and grime off the outside and use a nylon brush on the inside of the neck, if I cleaned them at all. Some folks claim that the shower cleaner Ka-Boom cleans brass pretty well, but I wasn't impressed. All of my cases end up in a tumbler sooner or later, depending on what propellent I used in them. With tumblers starting under $45, and used ones even cheaper, they are handy to have.