The method used is based on determination of ultrasonic phase velocities (time-of-flight measurements) made mainly in non- symmetric planes at oblique angles of incidence. We track the minima of the time-of-flight found in the (y, z)-scan of a quasi- monochromatic ultrasonic tone burst for each angle of incidence through a specimen. The specimen is mounted between a fixed transmitting transducer and a receiving transducer that is scanned in the (y, z) plane in a water filled immersion tank. The polar and azimuthal angles of incidence are varied by rotating the specimen. The data acquisition is controlled by a computer. From the measurements of polar and azimuthal angles of incidence, and time-of-flight in the immersion fluid, the direction cosines and corresponding phase velocities in the material can be determined. Using these values with an inversion routine for a general analytic solution of Christoffel Equation for wave propagation in an elastic material, we determine all the elastic constants of the material under test. Results from measurements for various materials are presented. The importance of the radiation pattern of transmitting transducer; initial alignment of the transducers and specimen are stressed. The minimum in the time-of-flight and maximum in the amplitude of the received ultrasonic signal must have the same (y, z) coordinates for a beam propagating in the x-direction to insure proper alignment.