Several short-period, zinc blende (CdTe)n/(MnTe)m superlattices, grown by molecular beam epitaxy, have been studied by Raman spectroscopy. All structures were deposited on (001) oriented GaAs or CdZnTe substrates. The folded longitudinal acoustic phonons (FLA), up to the fourth order, were well observed in the standard backscattering geometry with the incident and scattered polarization parallel to the <010> axis. The frequencies of those FLA correctly followed the usual dispersion law with the superlattice periodicity which allowed us to determine the unknown elastic constant C11 of zinc blende MnTe. The obtained value of C11 was compared to the value obtained by the extrapolation of results corresponding to CdMnTe mixed crystals. From the frequency position of the first order FLA doublet, the index of refraction of zinc blende MnTe was also obtained. Its value was compared with the value determined by a Kramers-Kronig analysis of the reflectivity spectrum, collected on thick MnTe layer.