  We study the motion of discrete interfaces driven by ferromagnetic
interactions in a two-dimensional periodic environment by coupling the
minimizing movements approach by Almgren, Taylor and Wang and a
discrete-to-continuous analysis. The case of a homogeneous environment has been
recently treated by Braides, Gelli and Novaga, showing that the effective
continuous motion is a flat motion related to the crystalline perimeter
obtained by $\Gamma$-convergence from the ferromagnetic energies, with an
additional discontinuous dependence on the curvature, giving in particular a
pinning threshold. In this paper we give an example showing that in general the
motion does not depend only on the $\Gamma$-limit, but also on geometrical
features that are not detected in the static description. In particular we show
how the pinning threshold is influenced by the microstructure and that the
effective motion is described by a new homogenized velocity.
