We explore the infrared spectrum of a three-dimensional dynamical model of planet HD 209458b as a function of orbital phase. The dynamical model predicts dayside atmospheric pressure-temperature profiles that are much more isothermal at pressures less than 1 bar than one-dimensional radiative-convective models have found. The resulting dayside thermal spectra are very similar to a blackbody, and only weak water absorption features are seen at short wavelengths. The dayside emission is consequently in better agreement with ground-based and space-based secondary eclipse data than any previous models, which predict strong flux peaks and deep absorption features. At other orbital phases, absorption due to carbon monoxide and methane is also predicted. We compute the spectra under two treatments of atmospheric chemistry: one uses the predictions of equilibrium chemistry, and the other uses nonequilibrium chemistry, which ties the timescales of methane and carbon monoxide chemistry to dynamical timescales. As a function of orbital phase, we predict planet-to-star flux ratios for standard infrared bands and all Spitzer Space Telescope bands. In Spitzer bands, we predict two- to fifteenfold variation in planetary flux as a function of orbital phase with equilibrium chemistry, and two- to fourfold variation with nonequilibrium chemistry. Variation is generally more pronounced in bands from 3 to 10 μm than at longer wavelengths. The orbital phase of maximum thermal emission in infrared bands is 15-45 orbital degrees before the time of secondary eclipse. Changes in flux as a function of orbital phase for HD 209458b should be observable with Spitzer, given the previously achieved observational error bars.