Summary: The relative communication performance of low- versus high-dimensional torus networks (\(k\)-ary \(n\)-cubes) has been extensively studied under various assumptions about communication patterns and technological constraints. In this paper, we extend the comparison to torus networks with incomplete, but regular, connectivities. Taking an \(n\)D torus as the basis, we show that a simple pruning scheme can be used to reduce the node degree from 2\(n\) to 4, while preserving many of the desirable properties of the intact network. Orienting the torus links (removing half of the channels) provides a second form of pruning that leads to (multidimensional) Manhattan street networks. Finally, combined pruning and orientation yields the fourth class of toroidal networks studied here. We compare the static performance parameters of these networks and evaluate their dynamic communication performance under the assumptions of virtual cut-through switching and constant pin count. The 3D case, leading to networks that are efficiently realizable with current technology, is used to demonstrate and quantify the performance benefits. Our results reinforce, extend, and complement previous studies that have demonstrated the performance advantages of low-dimensional \(k\)-ary \(n\)-cubes over higher-dimensional ones. For example pruned 3D tori provide additional design points that fall between 2D and 3D tori in terms of implementation complexity but can outperform both of these standard architectures. Thus, from a practical standpoint, pruning introduces additional flexibility in implementation options and trade-offs available to designers.