  This paper investigates the connection between the effective, large scale
behavior of Allen-Cahn energy functionals in periodic media and the sharp
interface limit of the associated $L^{2}$ gradient flows. By introducing a
Percival-type Lagrangian in the cylinder $\mathbb{R} \times \mathbb{T}^{d}$, we
establish a link between the $\Gamma$-convergence results of Anisini, Braides,
and Chiad\`{o} Piat and the sharp interface limit results of Barles and
Souganidis. In laminar media, we prove a sharp interface limit in a graphical
setting, making no assumptions other than sufficient smoothness of the
coefficients, and we prove that the effective interface velocity and surface
tension satisfy an Einstein relation. A number of pathologies are presented to
highlight difficulties that do not arise in the spatially homogeneous setting.
