Three-dimensional (3D) X-ray microscopy with submicron resolution has been used to make spatially resolved measurements of lattice curvature and elastic strain over two-dimensional slices in thin deformed Si plates. The techniques and capabilities associated with white-beam 3D X-ray microscopy are discussed, and both theoretical and experimental considerations associated with the measurement of Nye dislocation density tensors in deformed materials are presented. The ability to determine the local geometrically necessary dislocation (GND) density in the form of a dislocation density tensor, with micron spatial resolution over mesoscopic length scales, is demonstrated. Results are shown for the special case of an elastically bent (dislocation free) thin Si plate and for a similar thin Si plate that was bent plastically, above the brittle-to-ductile transition temperature, to introduce dislocations. Within the uncertainties of the measurements, the known result that GND density is zero for elastic bending is obtained, and well-defined GND distributions are observed in the plastically deformed Si plate. The direct and absolute connection between experimental measurements of GND density and multiscale modeling and computer simulations of deformation microstructures is discussed to highlight the importance of submicron-resolution 3D X-ray microscopy for mesoscale characterization of material defects and to achieve a fundamental understanding of deformation in ductile materials.