In response to growing interest in lightweight, high-strength wood-based engineering materials, a lattice core sandwich structure made of plywood and birch dowels with improved relative density in its core was designed and fabricated. Flatwise and edgewise compressive experiments were performed to investigate the mechanical behavior of the sandwich structure. The effect of relative density on the mechanical properties and failure mode of the structure under flatwise compression was discussed. The theoretical and experimental flatwise compression test results showed good agreement. The results of the edgewise compressive tests of the sandwich structure indicated that face sheet wrinkling, crushing, and macro-shear buckling of the core were the main failure modes. The wood-based lattice sandwich structure has potential applications in the construction industry as beam and plate structures for buildings.