We combine variability information from the MAssive Compact Halo Objects survey of the Large Magellanic Cloud with infrared photometry from the Spitzer Space Telescope Surveying the Agents of a Galaxy's Evolution survey to create a data set of ~30000 variable red sources. We photometrically classify these sources as being on the first ascent of the red giant branch, or as being in one of three stages along the asymptotic giant branch (AGB): oxygen-rich, carbon-rich, or highly reddened with indeterminate chemistry ("extreme" AGB candidates). We present linear period-luminosity (P-L) relationships for these sources using eight separate infrared bands (J, H, Ks, 3.6, 4.5, 5.8, 8.0, and 24um) as proxies for the luminosity. We find that the wavelength dependence of the slope of the P-L relationship is different for different photometrically determined classes of AGB stars. Stars photometrically classified as O-rich show the least variation of slope with wavelength, while dust enshrouded extreme AGB stars show a pronounced trend toward steeper slopes with increasing wavelength. We find that O-rich AGB stars pulsating in the fundamental mode obey a period-magnitude relation with a slope of -3.41+/-0.04 when magnitude is measured in the 3.6um band, in contrast to C-rich AGB stars, which obey a relation of slope -3.77+/-0.05.