The unified theory for elastic wave propagation in polycrystalline materials, developed by Stanke and Kino [J. Acoust. Soc. Am. 75, 665–681 (1984)], has been applied to compute the propagation constants in materials consisting of cubic crystallites with texture. The particular texture considered here is one in which one of the cube axes of each crystallite is aligned in a preferred direction with the other two being randomly oriented, leading to a material with macroscopic transverse isotropy. The calculations are done for plane waves under the assumption that the single crystal anisotropy is not large. Numerical results for the variations of attenuation and phase velocity with direction of propagation and frequency are presented for L, SH, and SVwaves.