Robust atmospheric and radiative transfer modeling will be required to properly interpret reflected-light and thermal emission spectra of terrestrial exoplanets. This will help break observational degeneracies between the numerous atmospheric, planetary, and stellar factors that drive planetary climate. Here, we simulate the climates of earthlike worlds around the Sun with increasingly slow rotation periods, from earthlike to fully Sun-synchronous, using the ROCKE-3D general circulation model. We then provide these results as input to the Spectral Planet Model, which employs the Spectral Mapping Atmospheric Radiative Transfer model to simulate the spectra of a planet as it would be observed from a future space-based telescope. We find that the primary observable effects of slowing planetary rotation rate are the altered cloud distributions, altitudes, and opacities that subsequently drive many changes to the spectra by altering the absorption band depths of biologically relevant gas species (e.g., ${{rm{H}}}_{2}{rm{O}}$, ${{rm{O}}}_{2}$, and ${{rm{O}}}_{3}$). We also identify a potentially diagnostic feature of synchronously rotating worlds in mid-infrared ${{rm{H}}}_{2}{rm{O}}$ absorption/emission lines.