We report tensorial tomographic Fourier ptychography (T2oFu), a nonscanning label-free tomographic microscopy method for simultaneous imaging of quantitative phase and anisotropic specimen information in 3D. Built upon Fourier ptychography, a quantitative phase imaging technique, T2oFu additionally highlights the vectorial nature of light. The imaging setup consists of a standard microscope equipped with an LED matrix, a polarization generator, and a polarization-sensitive camera. Permittivity tensors of anisotropic samples are computationally recovered from polarized intensity measurements across three dimensions. We demonstrate T2oFu’s efficiency through volumetric reconstructions of refractive index, birefringence, and orientation for various validation samples, as well as tissue samples from muscle fibers and diseased heart tissue. Our reconstructions of healthy muscle fibers reveal their 3D fine-filament structures with consistent orientations. Additionally, we demonstrate reconstructions of a heart tissue sample that carries important polarization information for detecting cardiac amyloidosis.