The theory of ultrasonic propagation in polycrystals with independent and uniformly distributed orientations of the grains presented in previous papers [J. Acoust. Soc. Am. 72, 1021–1031 (1982); 73, 1160–1163 (1983)] is generalized to calculate the scattering coefficients and the phase and group velocities of plane compressional and shear waves in textured polycrystals. The calculation was done for plane waves in polycrystals of cubic symmetry with rolling texture in second-order perturbation theory using the assumption that the changes in the material constants from grain to grain are small. In the limit texture equal to zero the analytical results are exactly the same as those for untextured polycrystals previously presented. Numerical calculations are carried out for some examples.