Eat the cherry as normal, separating the pit with your teeth and tongue as usual, but once the pit is free use your tongue to tuck it up under your upper lip. Once it's there, it will stay safely out of the way while you chew and swallow the rest of the cherry. You can repeat this several times; depending on your mouth geometry you can probably hold six to ten pits.

Asking around and looking online, it seems a lot of people don't know you can do this? I think this is because the pits are not especially visible, and the only clue someone is doing this is that they're needing to remove pits much less often.