Tensegrity structures have recently shown great potential as bracing devices for seismic control due to their unique ability to passively dissipate energy in structures subjected to severe deformations. Indeed, behaving as nonlinear springs, they are able to dissipate a great amount of energy energy during mechanical loading-unloading cycles. Planar tensegrity D-bar systems composed of four bars forming a rhombus, internally stabilized through a set of two perpendicular Shape Memory Alloy (SMA) cables, represent excellent candidates to act as braces for seismically resistant structures. However, although their tapered configuration maximizes in-plane buckling resistance with minimal mass, the out-of-plane buckling of such systems can compromise their overall structural efficiency, potentially engendering damage into adjacent non structural elements. In this paper, the efficiency of three-dimensional D-bar tensegrity structures under compressive loads is examined with the aim of proposing an advantageous design of D-bar-based bracing systems with optimized masses. We show that, by introducing a pre-strain in the superelastic cables, it is possible to achieve a wide shaped hysteresis, which yields to a significant amount of equivalent viscous damping (up to 30%). The presented numerical results on the energy dissipation properties of the examined structures, corroborated by experimental measurements of the buckling response, shed light on the research field of three-dimensional tensegrity structures, as efficient and lightweight bracing devices for seismic control.