This paper considers the multi-node multicast problem in a multi-dimensional wormhole-routed torus/mesh, where there is an arbitrary number of source nodes, each intending to multicast a message to an arbitrary set of destinations. This problem requires a large amount of bandwidth and, thus, typically incurs heavy contention and congestion. Evenly balancing the traffic load around the network is critical to achieving good performance. We show how to use a network-partitioning approach to achieve this goal. Simulation results show significant improvement over existing results in 3D tori and meshes. This work is an extension of our earlier work [22] from 2D tori/meshes to higher dimensional ones.