Due　to　its　good　potential　for　digital　signal　processing　analysis,　the　discrete　version　of　Gabor　analysis　has　interested　many　mathematicians.　In　this　paper,　we　focus　on　Gabor　analysis　on　discrete　periodic　sets,　which　can　model　a　signal　to　appear　periodically　but　intermittently.　For　a　Gabor　frame　on　a　discrete　periodic　set,　three　types　of　dual　frames　with　Gabor　structure　are　introduced　and　investigated　in　terms　of　discrete　Zak　transform.　Given　a　Gabor　frame　on　a　discrete　periodic　set,　a　characterization　of　a　Gabor　system　being　separately　its　dual　of　types　I-III　is　obtained.　A　method　to　construct　these　three　types　of　duals　is　presented.　An　explicit　expression　of　its　canonical　dual　is　established.　The　uniqueness　of　these　three　types　of　duals　is　characterized.　Some　examples　are　also　provided　to　illustrate　the　generality　of　the　theory.