Biological materials such as nacre have evolved microstructural design principles that result in outstanding mechanical properties. While nacre’s design concepts have led to bio-inspired materials with enhanced fracture toughness, the microstructural features underlying the remarkable damping properties of this biological material have not yet been fully explored in synthetic composites. Here, we study the damping behavior of nacre-like composites containing mineral bridges and platelet asperities as nanoscale structural features within its brick-and-mortar architecture. Dynamic mechanical analysis was performed to experimentally elucidate the role of these features on the damping response of the nacre-like composites. By enhancing stress transfer between platelets and at the brick/mortar interface, mineral bridges and nano-asperities were found to improve the damping performance of the composite to levels that surpass many biological and man-made materials. Surprisingly, the improved properties are achieved without reaching the perfect organization of the biological counterparts. Our nacre-like composites display a loss modulus 2.4-fold higher than natural nacre and 1.4-fold more than highly dissipative natural fiber composites. These findings shed light on the role of nanoscale structural features on the dynamic mechanical properties of nacre and offer design concepts for the manufacturing of bio-inspired composites for high-performance damping applications.