When a suspension of colloidal particles is placed in an oscillating electric field, the contrast in dielectric constant between the particles and the solvent induces a dipole moment in each of the colloidal particles. The resulting dipole-dipole interactions can strongly influence the phase behavior of the system. We investigate the phase behavior of cube-shaped colloidal particles in electric fields, using both experiments and Monte Carlo simulations. In addition to a string fluid phase and a body centered tetragonal (BCT) crystal phase, we observe a columnar phase consisting of hexagonally ordered strings of rotationally disordered cubes. By simulating the system for a range of pressures and electric field strengths, we map out the phase diagram, and compare the results to the experimentally observed phases. Additionally, we estimate the accuracy of a point-dipole approximation on the alignment of cubes in string-like clusters.