On a mesoscale the pore structure in natural rocks may have preferred orientation (texture) which leads to anisotropy of permeability, tortuosity, and shape factor. It is shown that such an anisotropic fluid saturated porous medium supports four different wave types: fast and slow quasilongitudinal and two quasishear waves. These results indicate that the velocities of the fast quasilongitudinal and two quasishear waves mostly depend on the properties of the frame and are not sensitive to the permeability and tortuosity directly (the frame stiffnesses, permeability, and tortuosity are indirectly related due to dependence on pore structure, however formally they can be considered as independent parameters). Thus by measuring these velocities one could determine frame elastic constants. The slow wave velocity, on the contrary, mostly depends on the pore geometry. Its angular dependence in a water- or air-saturated solid allows us to recover the components of the permeability and tortuosity tensors. This approach opens new challenges in determination of such characteristics of porous materials as preferred pore orientation and tortuosity which had been previously inaccessible experimentally.