I've played around with a lot of different cleaning methods over the years but the best combo that works for me is to use a rock tumbler with a simple water rinse afterwards. Find some sand or pebbles (something hard enough to slightly scratch copper, which is the prime ingredient in clad coinage), and then find some solid copper wire and cut it into 1/2" to 3/4" strands. I recently found a short section of solid copper braided cable in my yard and cut it up with some wire snips. I lost the original tumbling chamber, but I found that a half-gallon apple juice container fits the tumbler perfectly - but I have to change said container because the rocks and medium eventually get through the container walls! I also fill it 3/4 full of water and add a drop or two of Dawn soap.
Use maybe a handful of sand, and two handfuls of copper wire sections (1:2 ratio) for your tumbling medium. Tumble for 6-8 hours, then rinse. Clad coins come out of the solution with a slightly shiny, smooth surface, and picky cashiers won't refuse to take the coins. Works on copper pennies, nickels, clad dimes and quarters, coppery tokens, zincolns, bronze, and brass items. Seems to help the overall appearance of severely pitted copper pennies and Wheats that have lived underground for 60+ years.
Don't tumble them too long or you will end up with a bunch of blanks! This takes about 3-5 days of solid tumbling though. Usually 6-8 hours is plenty depending on the condition of the coins! Good luck!
I can post pics of my setup when I get home.