  We show decay estimates for the propagator of the discrete Schr\"odinger and
Klein-Gordon equations in the form $\norm{U(t)f}{l^\infty}\leq C
(1+|t|)^{-d/3}\norm{f}{l^1}$. This implies a corresponding (restricted) set of
Strichartz estimates. Applications of the latter include the existence of
excitation thresholds for certain regimes of the parameters and the decay of
small initial data for relevant $l^p$ norms. The analytical decay estimates are
corroborated with numerical results.
