The temperature dependence of the dynamic Young’s modulus E, the elastic energy dissipation coefficient Q−1 and the heat flow (DSC) has been studied between 20 and 370 K in a H-free and H-doped (nH=H/Me=0.006 and 0.01 at.) Ni30Ti50Cu20 alloy. The Young’s modulus exhibits softening when the start temperature Ms of the B2→B19 martensitic transition is approached on cooling and a steep modulus decrease between Ms and Mf. This steep decrease is associated with stress-induced motions of twin boundaries within the B19 martensite. Hydrogen reduces background damping of the martensite and dramatically enhances the dissipation in the temperature region of the transformation. These observations suggest that hydrogen (a) forms fixed pinning points for twin boundaries at low temperature and (b) gives rise to an anelastic relaxation PH associated with H dipoles and to a peak PAM due to H-twin boundary interactions.