  The Agmon estimate for multi-dimensional discrete Schr\"{o}dinger operators
is studied with emphasis on the microlocal analysis on the torus. We first
consider the semiclassical setting where semiclassical continuous
Schr\"{o}dinger operators are discretized with the mesh width proportional to
the semiclassical parameter. Under this setting, the Agmon estimate for
eigenfunctions is described by an Agmon metric, which is a Finsler metric
rather than a Riemannian metric. Klein-Rosenberger (2008) proved this by a
different argument in the case of a potential minimum. We also prove the Agmon
estimate and the optimal anisotropic exponential decay of eigenfunctions for
discrete Schr\"{o}dinger operators in the non-semiclassical standard setting.
