  We study spin systems on Bethe lattices constructed from d-dimensional
hypercubes. Although these lattices are not tree-like, and therefore closer to
real cubic lattices than Bethe lattices or regular random graphs, one can still
use the Bethe-Peierls method to derive exact equations for the magnetization
and other thermodynamic quantities. We compute phase diagrams for ferromagnetic
Ising models on hypercubic Bethe lattices with dimension d=2, 3, and 4. Our
results are in good agreement with the results of the same models on
d-dimensional cubic lattices, for low and high temperatures, and offer an
improvement over the conventional Bethe lattice with connectivity k=2d.
