In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in R^d. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R², giving a combinatorial characterization of graphs whose generic realizations in R² are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R² in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán.