Photonic crystals are widely known for their light-trapping capabilities. This is often associated with the occurrence of a photonic band gap or other suppression in the electromagnetic density of states. This also enables unprecedented forms of strong-coupling between light and matter. A less studied form of light-trapping occurs in the higher bands of a photonic crystal, where the electromagnetic density of states is enhanced rather than suppressed. This enables strong absorption of light from a broadband source over a wide acceptance angle in a material with weak intrinsic absorption. We describe designs of 3D photonic crystal silicon-based solar cells that enhance the overall absorption of sunlight using architectures consisting of less than 1 micron (equivalent bulk thickness) of silicon and no metallic mirrors. These crystals trap light through a parallel-to-interface negative refraction (PIR) effect that occurs over a broad angular and frequency range. These 3D photonic crystals exhibit an enhanced electromagnetic density of states, consisting of slow group velocity modes, in which the flow of energy is transverse to the depth of a thin film of material. In the case of a modulated nanowire photonic crystal solar cell, each wire contains a radial P-N junction and regions between the wires are filled with silica. This structure provides exceptional light absorption over a broad range of incident angles from 0 to 80 degrees. It is shown to absorb roughly 75% of all available sunlight in the wavelength range of 400-1000 nm. Power efficiencies in the range of 15-20% are shown using one micron of silicon. These nanostructured photonic crystals offer additional opportunities for solar spectral reshaping and novel electronic management to rival and possibly surpass the well-known Shockley-Queisser power efficiency limit.