We have utilized optical constants derived from thermal infrared spectra acquired by the Mariner 9 Infrared Interferometer Spectrometer (M9 IRIS) and Mars Global Surveyor Thermal Emission Spectrometer (MGS TES), during periods of relatively high dust/low water ice opacity, to generate synthetic transmission and emission spectra of the atmospheric dust. Using libraries of transmission and emission spectra of common rock-forming minerals and their alteration products, we applied a linear deconvolution algorithm to the spectra to model the dust mineralogy. Models of the M9 IRIS transmission data generally are of poor quality, and the mineralogical results of these fits are unlikely to be valid. Fits to the TES transmission spectrum and TES and IRIS emission spectra are of better quality and suggest that the dust mineralogy is dominated by framework silicates (feldspar and/or zeolite) with carbonate, sulfate, pyroxene, and olivine, identified near, but below, generalized detection limits. Evidence for phyllosilicates is not robust. Our results, combined with those from in situ measurements and other visible to infrared spectral studies, suggest that framework silicates (probably feldspar) dominate the dust mineralogy, with lesser amounts of olivine, pyroxene, amorphous material, hematite, and magnetite, and that the dust largely is the product of mechanical weathering of basaltic rocks with minor chemical alteration.