Network controllability is essential for designing the control algorithms of networked systems, among which the minimal controllability problem has attracted much attention of researchers due to the need of conserving control resources in practical applications. In this paper, we investigate the minimal controllability problem of Laplacian dynamics on two classes of non-planar graphs, namely sphere grid graphs and torus grid graphs. Since the Laplacian matrices of sphere grid graphs are closely related to the adjacency matrices of cylinder grid graphs, we first analyze the minimal numbers of control nodes required to guarantee controllability of networked linear systems whose system matrices are adjacency matrices of cylinder grid graphs. By exploring the relation between Laplacian matrices of sphere grid graphs and adjacency matrices of cylinder grid graphs, we then establish upper and lower bounds for the minimal number of control nodes required to guarantee controllability of Laplacian dynamic systems on sphere grid graphs. Finally, we provide the exact minimal number of control nodes and a corresponding control node set to guarantee controllability of the Laplacian dynamic systems on torus grid graphs.