The first steps toward measuring the elastic and piezoelectric constants of piezoelectric solids using resonant ultrasound spectroscopy (RUS) have been undertaken. Specifically, free-vibration frequencies of a piezoelectric LiNbO[inf 3] crystal were measured up to 1 MHz. The vibration frequencies of the piezoelectric parallelepiped were also predicted using the finite element method with published values of the elastic, piezoelectric, and dielectric constants of LiNbO[inf 3]. Published values of these constants from different sources using different methods are in reasonably good agreement. They also agree well with measurements of some of the constants performed in this study using a pulse-echo technique. Predicted piezoelectric vibration frequencies agree well with measured frequencies. Predictions from an elastic vibration analysis underestimate those from a piezoelectric vibration analysis by up to 12%. The results presented here suggest the feasibility of developing an inversion method to extract both the elastic and piezoelectric constants. This would extend the study of Ohno [I. Ohno, Phys. Chem. Minerals 17, 371--378 (1990)] who used RUS to determine the elastic constants of alpha quartz. The piezoelectric effect was considered, but no attempt was made to determine the piezoelectric constants as they were held fixed during the inversion to obtain the elastic constants.