For the comprehensive analysis of anisotropic materials, a new approach, called 'polarized light field microscopy' is introduced. It uses an LC-PolScope to which a microlens array was added at the image plane of the objective lens. The system is patterned after the 'light field microscope' that achieves both lateral and axial resolution in thick specimens in a single camera exposure. In polarized light field microscopy, the microlens array generates a hybrid image consisting of an array of small conoscopic images, each sampling a different object area. Analysis of the conoscopic images reveals the birefringence of each object area as a function of the propagation direction of transmitted light rays. The principles and utility of the instrument that we are calling 'light field LC-PolScope' are demonstrated with images of a thin, polycrystalline calcite film, revealing the azimuth and inclination angle of the optic axis for many crystals simultaneously, including crystals with diameters as small as 2 microm. Compared to traditional conoscopy and related methods, the vastly improved throughput and quantitative analysis afforded by the light field LC-PolScope make it the instrument of choice for measuring 3D birefringence parameters of complex structures.