Copper takes only a little more maintenance than sterling or other white/grey metals. Most of the time it comes down to simply cleaning more regularly. How often depends on the humidity of your climate, the chemistry of your skin, and how much you wear it. The quickest way is to give your jewelry a hand-scrubbing with mild soap and water. If you use a sink be sure to close the drain first, especially with earring designs. A soapy maille earring can slide down the drain lightning fast, like it is on ball bearings!
The easiest and most effective way to clean your jewelry, especially if you have a lot, is with a rotary rock tumbler. A high quality single-barrel tumbler and 2 lbs of stainless steel shot can be easily found for about $100. Cheap tumblers can go for as little as $30, while shot is usually around $16/lb. Be sure to only buy shot that is pin-free, as pins are used for getting into tight spots and aren’t necessary for maille, and may even be damaging. Never tumble pieces that have been coated or intentionally oxidized (bali silver) or you will remove their coating. Anodized niobium and enameled copper may be tumbled, but do it infrequently and only 5 - 20 minutes at a time.
Any solid metal jewelry without delicate beads or gemstones can be tumbled. Generally anything with a mohs hardness rating of 7 or higher *should* be okay, but if you're unsure it's best to err on the cautious side. Never tumble corundums (sapphires & rubies) as they are extremely brittle, regardless of their hardness rating of 9. If in doubt and if possible remove sensitive parts of a piece before tumbling.
Tumblers will polish most of your jewelry more completely than anything you can do with a cloth and paste, and although chemical cleaners like Brasso & TarnX are faster (seconds vs. minutes) they are expensive consumables, and harmful to our environment. These cleaners only really need to be used for completely stripping the piece or removing heavy tarnish. The only consumable needed with a tumbler is simple dish or hand soap, electricity and water. The traditional (detergent-free) blue colored Dawn is widely recommended and what I use for my pieces.

