I tried a bunch of home remedies like vinegar/salt, lemon juice/salt, baking soda, ketchup. They all worked somewhat, but for some reason the next day (I didn't leave them in the solution overnight - I just left them out after drying them) they looked awful again. Some just looked old and tarnished again, some were spotty, some had reddish marks, and a few had turned greenish.
Then I tried Mother's Mag and Aluminum Polish (use a small amount and brush lightly with a toothbrush, then wipe off residue with a clean cloth) and the pennies look amazing. Still do 2 days later. I got it from AutoZone, but I'm sure you can find it, or a similar metal polish, from any auto parts store.
For the record, I used all pre-1982 pennies so I don't get the zinc striations after pressing, so I don't know what the polish would do to a new penny, but I highly suggest using it on pre-1982 pennies. It's amazing how shiny and clean they are.