Since the anisotropy of ultrasonic SH wave velocities in plates was shown available to the nondestructive measurement of residual stress, much attention has been paid to the direction-dependence of propagation characteristics of ultrasonic waves. In this paper, it is intended to extend the acoustoelastic relation in slightly orthotropic materials to an arbitrary propagation direction of plane elastic waves. At first, the perturbation method was applied to the eigen value problem of the general acoustical tensor, and the formal solutions were obtained within the first-order of stress and the second-order of elastic anisotropy. Then these solutions were applied to the case where the effects of stress and anisotropy are of equal degree, and the velocities and polarization directions were explicitly expressed within the first-order of stress and anisotropy as the functions of the angles θ and φ specifying the propagation direction with respect to the principal axes of orthotropy. From these expressions, which are approximate but practically sufficient, the results previously obtained by Iwashimizu et al., King et al., and Thompson et al. were rederived by specializing the propagation direction. Finally, the simplification of the general relations was considered by imposing those restrictions on the anisotropic parameters, which are suggested by the theoretical analysis of macroscopic elastic constants of textured polycrystalline aggregates.