The association of experimental data showing that the plastic deformation of olivine, the main constituent of the upper mantle, is highly anisotropic and the ubiquitous seismic anisotropy in the upper mantle, which indicates that olivine crystals show coherent orientations over scales of tens to hundreds of kilometers, implies that the long-term deformation in the upper mantle is anisotropic. We propose a multiscale approach, based on a combination of finite element and homogenization techniques, to model the deformation of a lithospheric plate while fully considering the mechanical anisotropy stemming from a strain-induced orientation of olivine crystals in the mantle. This multiscale model explicitly takes into account the evolution of crystal preferred orientations (CPO) of olivine and of the mechanical anisotropy during the deformation. We performed a series of numerical experiments simulating the uniaxial extension of a homogeneous (100% olivine) but anisotropic plate to test the role of the olivine CPO on the plate mechanical behavior and the link between CPO and mechanical anisotropy evolution. Even for this simple solicitation, different orientations and intensity of the initial olivine CPO result in variable plate strengths and deformation regimes. A plate with an initial CPO where the olivine [100] and [010] axes are concentrated at 45 degrees to the extension direction has high resolved shear stresses on the easy (010)[100] and (001)[100] slip systems of olivine. This results in low strength and in deformation by transtension. Plates with an initial CPO where the maximum of [100] axes is parallel or normal to the extension direction show a high initial strength. Isotropic plates have an intermediate behavior. The progressive rotation of olivine [100] axes toward the imposed stretching direction results in hardening in all models, except in those characterized by an initial concentration of olivine [100] axes normal to the imposed extension, in which softening is followed by hardening.