Supergiant ionized filamentary structures with dimensions of up to almost-equal-to 2 kpc have been observed on deep H-alpha images of the irregular galaxies NGC 4449, an actively star-forming giant, and Sextans A, a lower surface brightness dwarf. These filaments are neither centered on nor in close proximity to luminous OB associations, and therefore do not fit the standard model for filament production from stellar-powered supershells. High-resolution echelle spectra for both galaxies demonstrate that filaments kinematically resemble typical extra-H II region ionized gas and generally have low, if any, expansion velocities. These observations and the large sizes of supergiant ionized filaments complicate shock models. Supergiant filaments could be photoionized, perhaps by leakage of Lyman-continuum radiation from classical H II complexes. Some supergiant filaments may be large-scale photoionization fronts on the surfaces of cool interstellar clouds. Such features could preferentially exist in galaxies, such as irregulars, where the interstellar medium is fairly transparent to Lyman-continuum radiation. Such a situation could also arise in systems in which the production of superbubbles has been high, leading to large regions of low H I density.