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Transverse bulk phonons in a multimode integrated quantum acoustic device are excited and characterized via their free-space coupling to a three-dimensional (3D) microwave cavity. These bulk acoustic modes are defined by the geometry of the Y-cut lithium niobate substrate in which they reside and couple to the cavity electric field via a large dipole antenna, with an interaction strength on the order of the cavity linewidth. Using finite element modeling, we determine that the bulk phonons excited by the cavity field have a transverse polarization with a shear velocity matching previously reported values. We demonstrate how the coupling between these transverse acoustic modes and the electric field of the 3D cavity depends on the relative orientation of the device dipole, with a coupling persisting to room temperature. Our study demonstrates the versatility of 3D microwave cavities for mediating contact-less coupling to quantum, and classical, piezoacoustic devices.