Three-dimensional Network-on-Chip (3-D NoC) is an emerging research topic exploring the network architecture of 3-D ICs that stack several smaller wafers for reducing wire length and wire delay. Although the network topology of 3-D NoC has been explored for a couple of years, there is still only a narrow range of choices. In this paper, we propose a class of 3-D topologies called Xbar-connected Network-on-Tiers (XNoTs), which consist of multiple network layers tightly connected via crossbar switches. To make the best use of the short delay and high density of inter-wafer links, XNoTs topologies have cross-bar switches that connect different layers and their cores. The planar topology on every layer can be independently customized so as to meet the cost-performance requirements, as far as network connectivity is at least guaranteed with the bottom layer. We also propose their routing algorithm, which guarantees deadlock-freedom by restricting the inter-layer packet transfer from a lower-numbered layer to a higher-numbered layer. Path sets at the bottom layer close to the heat sink of the chip can be selectively employed in order to mitigate the heat-dissipation problem of 3-D ICs. Several forms of XNoTs topologies including meshes, tori, and/or trees are created, and they are evaluated in terms of performance, cost, and energy consumption. As a result, we show that even with the flexibilities mentioned above, XNoTs achieve at least as high throughput as existing 3-D topologies for equivalent chip sizes.