  On the space of $\pm 1$ spin configurations on the 3$d$-square lattice, we
consider the \emph{shaken dynamics}, a parallel Markovian dynamics that can be
interpreted in terms of Probabilistic Cellular Automata. The transition
probabilities are defined in terms of pair ferromagnetic Ising-type
Hamiltonians with nearest neighbor interaction $J$, depending on an additional
parameter $q$, measuring the tendency of the system to remain locally in the
same state. Odd times and even times have different transition probabilities.
We compute the stationary measure of the shaken dynamics and we investigate its
relation with the Gibbs measure for the 3$d$ Ising model. It turns out that the
two parameters $J$ and $q$ tune the geometry of the underlying lattice. We
conjecture the existence of unique line of critical points in $J-q$ plane. By a
judicious use of perturbative methods we delimit the region where such curve
must lie and we perform numerical simulation to determine it. Our method allows
us to find in a unified way the critical values of $J$ for Ising model with
first neighbors interaction, defined on a whole class of lattices, intermediate
between the two-dimensional hexagonal and the three-dimensional cubic one, such
as, for example, the tetrahedral lattice. Finally we estimate the critical
exponents of the magnetic susceptibility and show that our model captures a
dimensional transition in the geometry of the system at $q = 0$.
