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Google Translate comes up with the phrase, Es ist dein Umzug, but I have a feeling that "Umzug" is the wrong word here. I think that "Zug" would be more appropriate.
The reason for this intuition is simply that "zug" is already found in a number of chess terms, e.g. "zwischenzug" for "in-between move" and "Zugzwang" for "compulsion to move".
Thus, I would be tempted to say Es ist dein Zug or dein Zug for short. Is my intuition correct?
Yes, "Zug" is correct in this context. While "es ist dein Zug" would be understood, "du bist am Zug" is more natural. Also natural is "du bist dran" (it's your turn).
Note that "Zug" only applies to board games, and not for example to card games.
Edit
"Dein Zug!" on its own is more natural than "es ist dein Zug", but the register is still "educated and old fashioned". Something you'd read in a book, not something you'd say today.
Trying to translate English literally ("It is your turn/move!", "Your turn/move!") usually just doesn't work.
You can use "du bist dran" for all games where the players take turns, and even in a metaphorical sense. So it's perfectly appropriate for chess or other board games.
As a native speaker I would recommend
Du bist dran!
As the most natural idiom.
Why? Because "Du bist am Zug" is more formally, suggesting a little bit that the other person should really move now, while "Du bist dran" rather suggests that the other person does something, while not being so much forced to move timely as if you say "Du bist am Zug".
Depends on how direct you want to say that its the other persons turn.
