Cold-rolled and annealed Ti50Ni40Cu10 shape memory alloy possesses a major (110)[001] texture along the rolling direction and a minor {111} γ-fiber texture along the normal direction. The damping capacity of the B2→B19 and B19→B2 martensitic transformation internal friction peaks for the Ti50Ni40Cu10 shape memory alloy was more pronounced in the rolling direction than in the transverse direction due to the effects of the cold-rolled and annealed textures. The damping capacity of the B19→B19’ and B19’→B19 martensitic transformation internal friction peaks was not noticeable affected by the orientation of the specimen