High-dispersion observations of the Na I D 5890, 5896 and K I 7665, 7699 interstellar lines, and the diffuse interstellar band at 5780 Angstroms in the spectra of 32 Type Ia supernovae are used as an independent means of probing dust extinction. We show that the dust extinction of the objects where the diffuse interstellar band at 5780 Angstroms is detected is consistent with the visual extinction derived from the supernova colors. This strongly suggests that the dust producing the extinction is predominantly located in the interstellar medium of the host galaxies and not in circumstellar material associated with the progenitor system. One quarter of the supernovae display anomalously large Na I column densities in comparison to the amount of dust extinction derived from their colors. Remarkably, all of the cases of unusually strong Na I D absorption correspond to "Blueshifted" profiles in the classification scheme of Sternberg et al. (2011). This coincidence suggests that outflowing circumstellar gas is responsible for at least some of the cases of anomalously large Na I column densities. Two supernovae with unusually strong Na I D absorption showed essentially normal K I column densities for the dust extinction implied by their colors, but this does not appear to be a universal characteristic. Overall, we find the most accurate predictor of individual supernova extinction to be the equivalent width of the diffuse interstellar band at 5780 Angstroms, and provide an empirical relation for its use. Finally, we identify ways of producing significant enhancements of the Na abundance of circumstellar material in both the single-degenerate and double-degenerate scenarios for the progenitor system. ; Comment: 55 pages, 13 figures, accepted for publication in ApJ