Radio observations of the Milky Way indicate that "filaments," or enlongated plasma structures, seen in synchrotron emission from accelerated electrons, are a common feature within our Galaxy. A number of theories attempt to explain this spatial structuring; however, we show that none can account quantitatively for the most striking aspect of these observations, namely, the observed transverse dimensions of the filaments. We then describe a new model, which provides both a plausible source of the accelerated electrons and a physical process that can explain the filamentation process. In particular, we show that a particle acceleration process, akin to the acceleration of the anomalous cosmic-ray component associated with the solar wind termination shock, can provide an economical explanation both for the acceleration and, in part, for the filamentation process; the filamentation process may then be further assisted by radiative instabilities driven by synchrotron emission. This model connects processes related to star formation with "activity" observed in the interstellar medium.