A downhole acoustic logging tool includes at least one acoustic transmitter and first and second arrays of acoustic receivers deployed on a tool body. The first array of acoustic receivers includes a plurality of axially spaced apart acoustic receivers and the second array of acoustic receivers includes a plurality of circumferentially spaced apart acoustic receivers. A method for making downhole acoustic logging measurements of a subterranean formation includes causing an acoustic logging tool to obtain axial acoustic logging measurements and circumferential acoustic logging measurements. The axial acoustic logging measurements and the circumferential acoustic logging measurements are then processed to compute each of five elastic constants of a transversely isotropic medium.