Shape memory alloys (SMAs) are well-known as high damping materials. Recently, we have reported that the Cu-Al-Mn SMAs show a high internal friction as well as the other SMAs such as Ni-Ti, Cu-Zn-Al SMAs. Since the CuAl-Mn-based SMAs possess an excellent cold-workability, a strong recrystallization texture can be formed by thermomechanical treatment. The present authors demonstrated that the texture control is considerably effective for improving the SM properties and especially, in the Cu-Al-Mn-Ni SMAs with a strong {112} recrystallization texture, a large pseudoelastic (PE) of about 7% can be obtained. Such a texture control is expected to enhance the damping capacity in the Cu-Al-Mn-based SMAs as well as PE. In this study, the effect of texture on the internal friction in Cu-Al-Mn and CuAl-Mn-Ni SMAs was investigated by tensile testing mode of Dynamic Mechanical Spectrometer (DMS). In sheet specimens with the {112} texture, value of dynamic damping tanψ depends on the loading direction and an excellent damping capacity of tanψ= 0. 1 can be obtained at the rolling direction in the martensite phase condition. Moreover, these Cu-Al-Mn-based SMAs show a high tensile strength over 600 MPa in the martensite phase. The present Cu-Al-Mn-based SMAs should have a great potential as the damping materials.