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. In the case of a homogeneous environment, recently treated by Braides, Gelli and Novaga, 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 a joint work with A. Braides, we show 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. In the last part of the talk, I would like to present also the results of an ongoing joint work with A. Braides: we use a discrete approximation of the motion by crystalline curvature to define an evolution of sets from a single point (nucleation) following a criterion of ``maximization'' of the perimeter, formally giving a backward version of the motion by crystalline curvature.