Years ago (before a lot of people on this board were born) I tried everything imaginable as tumbling media. I tried rice, strips of rags, sawdust, powdered Borax (don't try this one) and just about anything else I could get in the tumbler. The best of all of them is corn cob, with ground walnut shells next.
Sawdust will clean, but not really polish. It also leaves a lot of dust on the brass.
I currently buy a 40 pound bag of ground corn cob for $15.49 and add Berry's Brass Polish to it. I also use one tumbler (I have three tumblers) with plain corn cob to remove any polish residue when sizing in carbide dies, which does away with the fine scratching you get from the ultra fine grit of the polish. It doesn't matter what brand of brass polish you use, since they all act the same.
I've tried the car polish route, but it wasn't as satisfactory as the products made specifically for polishing brass cartridge cases. No matter what you use, avoid any polishes that contain ammonia, since it attacks the metal and will make it brittle. I've seen this happen and it ruins the brass.
Hope this helps.
Fred