We present a low-resolution (R = 90), 5.5-38 micron spectral sequence of a sample of M, L, and T dwarfs obtained with the Infrared Spectrograph (IRS) onboard the Spitzer Space Telescope. The spectra exhibit prominent absorption bands of H_2O at 6.27 microns, CH_4 at 7.65 microns, and NH_3 at 10.5 microns and are relatively featureless at lambda > 15 microns. Three spectral indices that measure the strengths of these bands are presented; H_2O absorption features are present throughout the MLT sequence while the CH_4 and NH_3 bands first appear at roughly the L/T transition. Although the spectra are, in general, qualitatively well matched by synthetic spectra that include the formation of spatially homogeneous silicate and iron condensate clouds, the spectra of the mid-type L dwarfs show an unexpected flattening from roughly 9 to 11 microns. We hypothesize that this may be a result of a population of small silicate grains that are not predicted in the cloud models. The spectrum of the peculiar T6 dwarf 2MASS J0937+2931 is suppressed from 5.5-7.5 microns relative to typical T6 dwarfs and may be a consequence of its mildly metal-poor/high surface gravity atmosphere. Finally, we compute bolometric luminosities of a subsample of the M, L, and T dwarfs by combining the IRS spectra with previously published 0.6-4.1 micron spectra and find good agreement with the values of Golimowski et al. who use L- and M-band photometry and to account for the flux emitted at lambda > 2.5 microns.