The sound velocities of longitudinal and shear waves are measured on a polycrystalline MgB2 superconductor with Tc of 39 K. The specimen used in the experiments is pressed and heated using the MgB2 powder. The elastic moduli, Debye temperature and specific heat at room temperature are obtained based on sound velocity data.The results indicate that the velocities are much higher than those in the usual materials, while elastic constants remain reasonably soft, which may be due to the high transition temperature of the MgB2 superconductor.