The uppermost layer of Earth's mantle, the asthenosphere, experiences large deformations due to a variety of tectonic processes. During deformation, grains of olivine, the main rock‐forming mineral in the asthenosphere, rotate into a preferred direction parallel to the deformation, developing a texture that can affect the response of the asthenosphere to tectonic stresses. Laboratory measurements show that the deformation rate depends on the orientation of the shear stress relative to the olivine texture. I will present numerical models that apply the findings of the laboratory measurements to geodynamic situations that are difficult to simulate in a laboratory. These models track the development of olivine texture and its directional response to shear stress, which are highly coupled. Our results suggest that anisotropic viscosity in the asthenosphere can significantly affect the motions of tectonic plates, as plate motion in a continuous direction should become faster, while abrupt changes in the direction of plate motion should meet high resistance in the underlying asthenosphere. We suggest that olivine textures in the asthenosphere play a critical role in upper mantle dynamics, that can have significant effects, for example, on plate motions.