We show clear experimental evidence of cooperative terahertz (THz) dynamics observed below 3 THz (∼100 cm^{-1}), for a low-symmetry Zr-based metal-organic framework structure, termed MIL-140A [ZrO(O_{2}C-C_{6}H_{4}-CO_{2})]. Utilizing a combination of high-resolution inelastic neutron scattering and synchrotron radiation far-infrared spectroscopy, we measured low-energy vibrations originating from the hindered rotations of organic linkers, whose energy barriers and detailed dynamics have been elucidated via ab initio density functional theory calculations. The complex pore architecture caused by the THz rotations has been characterized. We discovered an array of soft modes with trampolinelike motions, which could potentially be the source of anomalous mechanical phenomena such as negative thermal expansion. Our results demonstrate coordinated shear dynamics (2.47 THz), a mechanism which we have shown to destabilize the framework structure, in the exact crystallographic direction of the minimum shear modulus (G_{min}).