Let denote the d-dimensional grid with diagonals, that is, the graph with vertex set {1,2,…,n}^d and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary 2-coloring of the vertices of there exists a monochromatic connected subgraph with at least n^d−1−d²n^d−2 vertices. We also consider a d-dimensional triangulated grid; this is the graph of a triangulation of the solid cube [1,n]^d that refines the subdivision of [1,n]^d into the grid of unit cubes. Here every 2-coloring has a monochromatic connected subgraph with vertices.