A new class of truss structure based on superelastic shape memory alloy (SMA) wire has been developed by weaving superelastic SMA wire through two perforated facesheets. A gap was maintained between the facesheets while weaving and the ends of wire forming the truss legs are anchored in each facesheet. The resulting structure has a modified pyramidal configuration and is capable of undergoing large recoverable deformations typical of superelastic SMA. A four-unit cell truss specimen has been tested under static load cycles to investigate the compressive response. The truss specimen underwent a hysteretic loop and demonstrated minimal permanent deformation closely resembling the behavior of bulk SMA. A finite element model of the truss was generated and the analysis results were compared with the experimental response. The present work is an attempt to demonstrate an SMA-based truss structure having energy absorption capabilities with minimum permanent deformation. These truss structures may be applied for damage mitigation in composites subjected to impact and blast loads.