The mechanical properties of metal-organic frameworks (MOFs) are of high fundamental and practical relevance. A particularly intriguing technique for determining anisotropic elastic tensors is Brillouin scattering, which so far has rarely been used for highly complex materials like MOFs. In the present contribution, we apply this technique to study a newly synthesized MOF-type material, referred to as GUT2. The experiments are combined with state-of-the-art simulations of elastic properties and phonon bands, which are based on machine-learning force fields and dispersion-corrected density functional theory. This provides a comprehensive understanding of the experimental signals, which can be correlated to the longitudinal and transverse sound velocities of the material. Notably, the combination of the insights from simulations and experiments allows the determination of approximate values for the components of the elastic tensor of the studied material even when dealing with comparably small single crystals, which limit the range of accessible experimental data.