Solairaju and Chithra introduced a new type of labeling of a graph G with p vertices and q edges called an edge odd graceful labeling if there is a bijection f from the edges of the graph to the set {1, 3, ... , 2q - 1} such that, when each vertex is assigned the sum of all edges incident to it mod 2q, the resulting vertex labels are distinct. In this paper, we proved necessary and sufficient conditions for the cylinder grid graph Cm,n = Pm × Cn and torus grid graph Tm,n = Cm × Cn are edge odd graceful.