Anisotropy in geological materials has various sources. Most important is anisotropy related to lattice (crystallographic) preferred orientation (LPO), due to the anisotropy of rock-forming minerals and the texture of the polycrystalline aggregate. In addition, oriented microcracks, grain shape preferred orientation (SPO), and the presence and distribution of partial melt or fluids may also be of importance. Velocities of elastic waves propagating through a rock with oriented fractures are much higher parallel to the fractures than across them. With increasing pressure microfractures close and their contribution to elastic anisotropy diminishes. The remaining velocity anisotropy is nearly pressure-independent, and largely caused by the LPO (texture) of the constituent minerals. This paper reports results of experimental investigations and numerical calculations based on the LPOs of important rock-forming minerals and investigates the relationship between oriented cracks, texture and elastic rock properties such as velocity anisotropy, acoustic birefringence (shear wave splitting) and shear wave polarisation. Finally, the significance of LPO in Earth science investigations is briefly discussed.