Viscoelastic materials are commonly used to dissipate kinetic energy in case of impact and vibrations. Unfortunately, dissipating large amounts of energy in a monolithic material requires high combinations of two intrinsic properties – Young’s modulus and loss factor, which are generally in conflict. This limitation can be overcome by designing cellular materials incorporating negative stiffness elements. Here we investigate a configuration comprising two positive stiffness elements and one negative stiffness element. This unit cell possesses an internal degree of freedom, which introduces hysteresis under a loading-unloading cycle, resulting in substantial energy dissipation, while maintaining stiffness. We demonstrate and optimize a simple implementation in a single material design that does not require external stabilization or pre-compression of buckled elements; these key features make it amenable to fabrication by virtually any additive manufacturing approach (from 3D printing to assembly and brazing) in a wide range of base materials (from polymers to metals). No additional intrinsic damping mechanism is required for the base material, which is assumed linear elastic. Furthermore, the architected material can be designed to be fully recoverable. When optimized, these architected materials exhibit extremely high combinations of Young’s modulus and damping, far superior to those of each constituent phase.