Dwarf galaxies play a crucial role in our understanding of the formation and evolution of galaxies, and the concept of supernova-driven mass outflows is a vital ingredient in theories of the structure and evolution of dwarf galaxies. Despite the theoretical importance of these outflows, there is a very limited amount of direct observational evidence for their existence. We have therefore begun a detailed multi-wave-band search for outflows in dwarf (M(B) greater than or equal to -18) galaxies with extensive recent or ongoing centrally concentrated star formation. We report the first results of this search in the present paper. Observations of the ionized gas in dwarf amorphous galaxies with centrally concentrated populations of massive stars provide evidence for the large-scale expansion of their ionized interstellar media. Fabry-Perot H alpha images reveal the presence of kiloparsec-scale ''superbubbles'' and filaments which tend to be oriented along the galaxy minor axis. These structures are comparable in size to the characteristic optical sizes of the galaxies, and dominate the morphology of the galaxies at low surface brightness in H alpha. Echelle spectra taken through these structures demonstrate that the kinematics are consistent with hollow kiloparsec-scale structures expanding at typical velocities of about 50 km s(-1) (ranging from about 25 and 100 km s(-1)). The mechanical energy output from the supernovae and stellar winds in the central region of each galaxy appears adequate to power expansion motions of this size and speed. Since the inferred expansion velocities are comparable to (but probably a bit smaller than) the rather uncertain escape velocities from these galaxies, the ultimate fate of the accelerated gas is not clear. However, even if this relatively cool and dense H alpha-emitting gas is not ejected from the galaxies, it is likely that the much hotter (supernova and stellar wind heated) gas in the interior of the superbubble will escape the galaxies once the superbubble fragments. Since this hot gas contains the newly synthesized metals, the chemical evolution of dwarf galaxies may be strongly affected by these outflows, even if the galaxy is able to retain most of its interstellar medium. Since expanding structures of this size and velocity are not observed in all low-mass galaxies with recent or ongoing star formation, we suggest that we are witnessing transient events that likely have a relatively low ''duty cycle'' in such galaxies. That is, we argue that the particular galaxies in the present paper have had significantly elevated star formation rates over the past 10(7)-10(8) yr (i.e., these are starburst or young poststarburst systems). This interpretation is consistent with the optical colors and emission-line properties of these galaxies.