A six-dimensional symplectic tracking approach exploiting the periodicity properties of dielectric laser acceleration (DLA) gratings is presented. The longitudinal kick is obtained from the spatial Fourier harmonics of the laser field within the structure, and the transverse kicks are obtained using the Panofsky-Wenzel theorem. Additionally to the usual, strictly longitudinally periodic gratings, our approach is also applicable to periodicity chirped (subrelativistic) and tilted (deflection) gratings. In the limit of small kicks and short periods we obtain the 6D Hamiltonian, which allows, for example, to obtain matched beam distributions in DLAs. The scheme is applied to beam and grating parameters similar to recently performed experiments. The paper concludes with an outlook to laser based focusing schemes, which are promising to overcome fundamental interaction length limitations, in order to build an entire microchip-sized laser driven accelerator.