Supermassive black holes (SMBHs) are thought to provide energy that prevents catastrophic cooling in the centers of massive galaxies and galaxy clusters. However, it remains unclear how this "feedback" process operates. We use high-resolution optical data to study the kinematics of multiphase filamentary structures by measuring the velocity structure function (VSF) of the filaments over a wide range of scales in the centers of three nearby galaxy clusters: Perseus, A2597, and Virgo. We find that the motions of the filaments are turbulent in all three clusters studied. There is a clear correlation between features of the VSFs and the sizes of bubbles inflated by SMBH-driven jets. Our study demonstrates that SMBHs are the main driver of turbulent gas motions in the centers of relaxed galaxy clusters and suggests that this turbulence is an important channel for coupling feedback to the environment. Our measured amplitude of turbulence is in good agreement with Hitomi Doppler line broadening measurement and X-ray surface-brightness fluctuation analysis, suggesting that the motion of the cold filaments is well-coupled to that of the hot gas. The smallest scales that we probe are comparable to the mean free path in the intracluster medium. Our direct detection of turbulence on these scales provides the clearest evidence to date that isotropic viscosity is suppressed in the weakly collisional, magnetized intracluster plasma.