In nonlinear periodic media of arbitrary dimension we consider the small amplitude asymptotics of wavepackets. The wavepackets have carrier Bloch waves of equal frequency. We use the cubic Gross-Pitaevskii equation (GP) as a prototype of the governing equation. We discuss two classical asymptotic scalings, one (for ) leading to the nonlinear Schroedinger equation and one (for ) leading to first order coupled mode equations (CMEs) as effective amplitude equations. Both of these models can support solitary waves - thus predicting nearly solitary waves of the GP. In particular, the CMEs for for the case of the coupling of two counter-propagating Bloch waves support a family of solitary waves parametrized by the velocity . Can this be generalized to dimensions such that in the CMEs a solitary wave family parametrized by exists? Solitary waves are typically found in spectral gaps. For at least four () carrier waves are needed to produce CMEs with a spectral gap that supports solitary waves. However, only standing solitary waves have been found so far. We also provide a validity result of the dimensional NLS-asymptotics as well as the CME-asymptotics over asymptotically large time intervals.