A new approach for using acoustic measurements to evaluate residual stresses in the presence of unknown material property variations is presented. It is shown that measurements using shear waves propagating along the normal to the surface of a plate do not provide sufficient information to separate the influences of stress and material property variations. To overcome this fundamental limitation, an alternative theory is developed that governs the propagation of shear waves polarized horizontally with respect to the surface of a plate (SH-waves), but propagating at oblique angles with respect to the surface normal. The question of separating the effects of residual stress and material properties on acoustic velocity is addressed in detail. In addition, a practical experimental procedure is developed that permits the evalution of the in-plane components of the principal stresses in a plate exhibiting an unknown inhomogeneous initial anistropy caused by material texture or microstructure. The procedure is then verified experimentally using an aluminum specimen with a known residual stress state, but unknown initial anisotropy.