@emily I would probably call that one "turn" and have trouble naming the single thing (unless "move" was unambiguous). ^^*

"Turn cycle" is relatively good, if somewhat unwieldy, would probably work as long as it doesn't have to be referred to on parts of the game other than the manual (cards etc). Depending on the theming of the game picking a topic related name can also be a very good idea ("era", "generation", "episode"...).