This study is devoted to deducing exact elastic constants of an anisotropic solid material without using any advance information on the elastic constants by incorporating a displacement-distribution measurement into resonant ultrasound spectroscopy (RUS). The usual RUS method measures free-vibration resonance frequencies of a solid and compares them with calculations to find the most suitable set of elastic constants by an inverse calculation. This comparison requires mode identification for the measured resonance frequencies, which has been difficult and never been free from ambiguity. This study then adopts a laser-Doppler interferometer to measure the displacement-distribution patterns on a surface of the vibrating specimen mounted on pinducers; comparison of the measured displacement distributions with those computed permits us to correctly identify the measured resonance frequencies, leading to unmistakable determination of elastic constants. Because the displacement patterns are hardly affected by the elastic constants, an exact answer is surely obtained even when unreasonable elastic constants are used as initial guesses at the beginning of the inverse calculation. The usefulness of the present technique is demonstrated with an aluminum alloy and a langasite crystal.