The fate of ionizing radiation in starburst galaxies is key to understanding cosmic reionization. However, the galactic parameters on which the escape fraction of ionizing radiation depend are not well understood. Ionization-parameter mapping provides a simple, yet effective, way to study the radiative transfer in starburst galaxies. We obtain emission-line ratio maps of [SIII]/[SII] for six, nearby, dwarf starbursts: NGC 178, NGC 1482, NGC 1705, NGC 3125, NGC 7126, and He 2-10. The narrow-band images are obtained with the Maryland-Magellan Tunable Filter at Las Campanas Observatory. Using these data, we previously reported the discovery of an optically thin ionization cone in NGC 5253, and here we also discover a similar ionization cone in NGC 3125. This latter cone has an opening angle of 40+/-5 degrees (0.4 ster), indicating that the passageways through which ionizing radiation may travel correspond to a small solid angle. Additionally, there are three sample galaxies that have winds and/or superbubble activity, which should be conducive to escaping radiation, yet they are optically thick. These results support the scenario that an orientation bias limits our ability to directly detect escaping Lyman continuum in many starburst galaxies. A comparison of the star-formation properties and histories of the optically thin and thick galaxies is consistent with the model that high escape fractions are limited to galaxies that are old enough (> 3 Myr) for mechanical feedback to have cleared optically thin passageways in the ISM, but young enough (