We present a polarization-insensitive plasmonic photoconductive terahertz emitter that uses a two-dimensional array of nanoscale cross-shaped apertures as the plasmonic contact electrodes. The geometry of the cross-shaped apertures is set to maximize optical pump absorption in close proximity to the contact electrodes. The two-dimensional symmetry of the cross-shaped apertures offers a polarization-insensitive interaction between the plasmonic contact electrodes and optical pump beam. We experimentally demonstrate a polarization-insensitive terahertz radiation from the presented emitter in response to a femtosecond optical pump beam and similar terahertz radiation powers compared to previously demonstrated polarization-sensitive photoconductive emitters with plasmonic contact electrode gratings at the optimum optical pump polarization.