The experimental setup for precision acoustic measurements of the elastic tensor of structurally inhomogeneous materials has been designed and fabricated. The achieved accuracy of measurements is high enough for studying a fine structure of rocks. It is shown that resonant acoustic spectroscopy is a useful tool for measuring the small content of oriented microcracks that can be used for diagnostics of early stages of fracture when oriented microcracking occurs. Elastic characteristics of a Westerly granite sample have been investigated. The elastic tensor has been found to be hexagonal, oriented with respect to the rift plane. The obtained data have shown the presence of oriented microcracks and have been explained sensibly by a model of microcracks.