Chromatin organization and dynamics are critical for gene regulation. In this work we present a methodology for fast and parallel three-dimensional (3D) tracking of multiple chromosomal loci of choice over many thousands of frames on various timescales. We achieved this by developing and combining fluorogenic and replenishable nanobody arrays, engineered point spread functions, and light sheet illumination. The result is gentle live-cell 3D tracking with excellent spatiotemporal resolution throughout the mammalian cell nucleus. Correction for both sample drift and nuclear translation facilitated accurate long-term tracking of the chromatin dynamics. We demonstrate tracking both of fast dynamics (50 Hz) and over timescales extending to several hours, and we find both large heterogeneity between cells and apparent anisotropy in the dynamics in the axial direction. We further quantify the effect of inhibiting actin polymerization on the dynamics and find an overall increase in both the apparent diffusion coefficient D* and anomalous diffusion exponent α and a transition to more-isotropic dynamics in 3D after such treatment. We think that in the future our methodology will allow researchers to obtain a better fundamental understanding of chromatin dynamics and how it is altered during disease progression and after perturbations of cellular function.