  We explore the phase diagram of Ising spins on one-dimensional chains which
criss-cross in two perpendicular directions and which are connected by
interchain couplings. This system is of interest as a simpler, classical analog
of a quantum Hamiltonian which has been proposed as a model of magnetic
behavior in Nb$_{12}$O$_{29}$ and also, conceptually, as a geometry which is
intermediate between one and two dimensions. Using mean field theory as well as
Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively
the boundaries of four ordered phases. Each becomes an effective Ising model
with unique effective couplings at large interchain coupling. Away from this
limit we demonstrate non-trivial critical behavior, including tricritical
points which separate first and second order phase transitions. Finally, we
present evidence that this model belongs to the 2D Ising universality class.
