Using the finite element method, we analyzed the resonance properties of a longitudinal leaky surface acoustic wave (LLSAW) on the structure of a piezoelectric LiTaO3 (LT) or LiNbO3 (LN) thin plate bonded to a diamond support substrate. When the plate thickness was 0.3 wavelength or larger, the particle displacements of the LLSAW were substantially concentrated near the surface, and the resonance properties were significantly improved, compared to those of the single LT or LN. However, spurious responses due to other SAW modes also emerged. Regarding the LN thin plate, we obtained a phase velocity and fractional bandwidth of approximately 7300 m/s and 9.4%, respectively. Furthermore, a structure was proposed to suppress spurious responses, in which the piezoelectric thin plate is divided into two layers with different Euler angles. The spurious responses were significantly suppressed after dividing the thin plate, while the main response of the LLSAW was maintained.