We investigate the damping enhancement in a class of biomimetic staggered composites via a combination of design, modeling, and experiment. In total, three kinds of staggered composites are designed by mimicking the structure of bone and nacre. These composite designs are realized by 3D printing a rigid plastic and a viscous elastomer simultaneously. Greatly-enhanced energy dissipation in the designed composites is observed from both the experimental results and theoretical prediction. The designed polymer composites have loss modulus up to ~500 MPa, higher than most of the existing polymers. In addition, their specific loss modulus (up to 0.43 km²/s²) is among the highest of damping materials. The damping enhancement is attributed to the large shear deformation of the viscous soft matrix and the large strengthening effect from the rigid inclusion phase.