The dielectric tensor is a physical descriptor of fundamental light-matter interactions, characterizing anisotropic materials with principal refractive indices and optic axes. Despite its importance in scientific and industrial applications ranging from material science to soft matter physics, the direct measurement of the three-dimensional dielectric tensor has been limited by the vectorial and inhomogeneous nature of light scattering from anisotropic materials. Here, we present a dielectric tensor tomographic approach to directly measure dielectric tensors of anisotropic structures including the spatial variations of principal refractive indices and directors. The anisotropic structure is illuminated with a polarized plane wave with various angles and polarization states. Then, the scattered fields are holographically measured and converted into vectorial diffracted field components. Finally, by inversely solving a vectorial wave equation, the three-dimensional dielectric tensor is reconstructed. Using this approach, we demonstrate quantitative tomographic measurements of various nematic liquid-crystal structures and their fast three-dimensional non-equilibrium dynamics. Measuring three-dimensional dielectric tensors is desired for applications in material and soft matter physics. Here, the authors use a tomographic approach and inversely solve the vectorial wave equation to directly reconstruct dielectric tensors of anisotropic structures.