  In this paper we investigate the long-term behaviour of solutions to the
discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that
front-like initial conditions evolve towards a planar travelling wave modulated
by a phaseshift $\gamma_l(t)$ that depends on the coordinate $l$ transverse to
the primary direction of propagation. This direction is allowed to be general,
but rational, generalizing earlier known results for the horizontal direction.
We show that the behaviour of $\gamma$ can be asymptotically linked to the
behaviour of a suitably discretized mean curvature flow. This allows us to show
that travelling waves propagating in rational directions are nonlinearly stable
with respect to perturbations that are asymptotically periodic in the
transverse direction.
