A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted γ ×k,t (G). We give a Vizing-like inequality for Cartesian product graphs, namelywhere ρ is the packing number. We also give bounds on γ ×k,t (G H) in terms of (open) packing numbers, and consider the extremal case of γ ×k,t (Kn Km), i.e., the rook's graph, giving a constructive proof of a general formula for γ×2,t(Kn Km).