Elastic constants of solids were, until recently, evaluated predominantly by pulse-echo ultrasonic methods which are based on measuring the speed of ultrasonic waves propagation in solids. Resonant ultrasound spectroscopy (RUS) is a relatively novel method in which all components of elastic tensor are determined from measured resonance frequencies of a freely vibrating specimen. The RUS technique has been employed in this work to investigate temperature dependence of the elastic properties of the parent austenite phase in CuAlNi shape memory alloy single crystals. This phase exhibits very high elastic anisotropy (anisotropy factor A 12) and softening the shear coefficient C0 upon cooling when approaching the Ms transformation temperature. The complications (need for large number of resonant frequencies) emerging when one tries to determine all elastic constants of highly elastically anisotropic materials by the RUS technique are discussed. It is concluded that the shear coefficients C0 and C44, which are the most important for shape memory alloys, are, nevertheless, determined reliably.