This paper presents the modeling and design exploration of tensegrity plate mechanisms with the ability of dissipating energy arising from compressive loads. The tensegrity plates are comprised of an assembly of tensegrity prisms, each formed by three compressive bars self-equilibrated by a network of tensile strings. The plates transfer a uniform compressive surface load applied along their planform area to uniaxial tension and compression within their members. The energy dissipation capabilities of plates with strings formed by three different elastoplastic metals and a pseudoelastic shape memory alloy (SMA) are explored. The constitutive parameters of these materials are calibrated from experimental data, and finite element models of the plates are created. A Taguchi design of experiments is used to evaluate the main effects of different design parameters of the plates on their energy dissipation and residual deformation responses. Results indicate that plates of larger thickness, lower diameter, and higher complexities provide higher energy dissipation per unit mass. Pseudoelastic SMA strings were the only type of strings that provided cyclic energy dissipation without the emergence of residual displacements. The studied energy absorbing mechanisms can potentially be integrated in aerospace, automotive, and civil components designed to absorb and dissipate energy from vibrations or distributed compressive loads.