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This article presents experimental, theoretical, and numerical studies of the propagation of guided ultrasonic waves in a layered epitaxial structure of garnet compounds. A microscopic model, which yields dispersion equations based on material and geometrical properties, is developed. Acoustic microscopy experiments on a YAG:Ce crystal substrate and an epitaxial structure containing LuAG:Ce single crystalline films, grown using the liquid phase epitaxy growth method onto a YAG:Ce crystal substrate, reveal distinct phase velocity behaviors. The YAG substrate exhibits consistent velocities, minimally influenced by frequency, while the epitaxial structure shows dispersion, indicating frequency-dependent phase velocities. Experimental results are compared with numerically calculated dispersion curves, showing high agreement in the low-frequency range and minor deviations at higher frequencies. An optimization procedure is developed and applied, starting with the YAG substrate and extending to the LuAG:Ce film/YAG:Ce crystal epitaxial structure. The procedure allows for the extraction of material properties, offering valuable insights into the mechanical characteristics of the all-solid-state LuAG:Ce film/YAG:Ce crystal structure. This research represents a significant advancement in understanding ultrasonic wave dynamics in layered structures, particularly unveiling previously unexplored elastic properties of LuAG:Ce single crystalline films as a well-known scintillation material.