We forecast the sensitivity with which the Murchison Widefield Array (MWA) can measure the 21 cm power spectrum of cosmic hydrogen, using radiative transfer simulations to model reionization and the 21 cm signal. The MWA is sensitive to roughly a decade in scale (wavenumbers of k ~ 0.1 - 1 h Mpc^{-1}), with foreground contamination precluding measurements on larger scales, and thermal detector noise limiting the small scale sensitivity. This amounts primarily to constraints on two numbers: the amplitude and slope of the 21 cm power spectrum on the scales probed. We find, however, that the redshift evolution in these quantities can yield important information about reionization. Although the power spectrum differs substantially across plausible models, a generic prediction is that the amplitude of the 21 cm power spectrum on MWA scales peaks near the epoch when the intergalactic medium (IGM) is ~ 50% ionized. Moreover, the slope of the 21 cm power spectrum on MWA scales flattens as the ionization fraction increases and the sizes of the HII regions grow. Considering detection sensitivity, we show that the optimal MWA antenna configuration for power spectrum measurements would pack all 500 antenna tiles as close as possible in a compact core. The MWA is sensitive enough in its optimal configuration to measure redshift evolution in the slope and amplitude of the 21 cm power spectrum. Detecting the characteristic redshift evolution of our models will confirm that observed 21 cm fluctuations originate from the IGM, and not from foregrounds, and provide an indirect constraint on the volume-filling factor of HII regions during reionization. After two years of observations under favorable conditions, the MWA can constrain the filling factor at an epoch when ~ 0.5 to within roughly +/- 0.1 at 2-sigma. ; Comment: 14 pages, 9 figures, submitted to ApJ