A laser-pumped accelerator utilizing a resonant, periodic, dielectric structure is proposed. The electromagnetic fields due to a side-injected laser beam impinging on this structure are calculated in a two-dimensional standing-wave approximation, leading to an estimated accelerating field in excess of 1 GV/m for accessible experimental parameters. The longitudinal dynamics of injected electrons in the device are discussed, as are the first-and second-order transverse focusing effects inherent to this structure. Similarities and differences between this scheme and conventional and other advanced accelerators are examined.