Two very important factors which determine the effectiveness of a pump are its volumetric and power efficiencies. Yin and Ghoneim constructed a prototype of a Flexible-Matrix-Composite (FMC) body pump with a very high volumetric efficiency or pumping potential (the relative volume reduction due to a relative input stroke). The high volumetric efficiency is attributed to the geometry of the pump’s structure (hyperboloid) as well as the high negative effective Poisson’s ratio of the 3-layer [θ/β/θ] flexible-matrix-composite (carbon/polyurethane) laminate adopted for the body of the pump. However, the power efficiency of the pump was not evaluated. It is the objective of the current paper to obtain an estimate of the power efficiency of the pump. The viscoelastic properties of the 3-layer FMC (carbon/polyurethane) laminate are evaluated experimentally using the Dynamic Mechanical Analyzer (DMA) as well as analytically by applying the correspondence principle together with the micro-mechanics approach. In order to obtain an estimate of the power efficiency of the FMC body pump, the axial and shear loss factors of a laminated infinitely long cylindrical tube as functions of β and θ fiber orientation angles are determined employing the Adam and Bacon approach. The analysis engenders high loss factors (greater than 0.4), which suggests that the power efficiency of the proposed pump using the 3-layer carbon/polyurethane laminate may be low.