Overcast has functionality corresponding to "mark as read" for episodes, but it doesn't call it that. It can dispense with this terminology, for now, because it doesn't stream episodes; an episode is either downloaded or not downloaded.
[Update, Oct 2016: streaming has long since been added, see below. Streamed episodes behave almost exactly like downloaded ones: you don’t “mark as read”, you “delete”.]
You can swipe an episode to the right to reveal a "Delete" button; pressing it deletes the downloaded copy of the episode. (You can still see the episode by going to the podcast and selecting All to see all the episodes in the feed.)
Overcast does remember which episodes you have downloaded before, and refrains from re-downloading episodes you have deleted. (you can of course manually download them again.)