Structural materials combining high stiffness and damping capabilities are in high demand for passive damping applications in vibration control, precision manufacturing, and resilient buildings. However, the development of enhanced passive damping materials with high stiffness at low weight has been hindered by the mutually excluding nature of these mechanical properties. Motivated by the mechanical performance of biological materials such as bone and nacre, we exploit a simple casting and magnetically assisted manufacturing process to fabricate platelet-reinforced polymers with a stiff and damping staggered architecture. Dynamic mechanical analysis of our staggered architectures reveals an increase in the stiffness of the composites by a factor up to 4.5 while maintaining the strong damping response of the host polymer. Using an established micromechanical model, we can predict the damping figure of merit of such composites and provide guidelines for the creation of stiff and damping bio-inspired structures while considering boundary conditions of the manufacturing process. By reaching mechanical performance superior or comparable to bone and nacre, our bio-inspired strategy proves to be a promising pathway for the further development of low-power passive damping materials.