We report spectroscopic redshifts for 18 microJy-radio galaxies at mean redshift of z=2.2 that are faint at both submmillimeter (submm) and optical wavelengths. While the radio fluxes of these galaxies could indicate far-infrared (far-IR) luminosities comparable to high-redshift submillimeter-selected galaxies (>10^12 Lsun), none are detected in the submm. We propose that this new population of galaxies represents an extension of the high-redshift submm galaxy population, but with hotter characteristic dust temperatures that shift the peak of their far-IR emission to shorter wavelengths, reducing the submm flux below the sensitivity of current instruments. Therefore, surveys in the submm waveband may miss up to half of the most luminous, dusty galaxies at z~2. Mid-infrared observations with Spitzer will be a powerful tool to test this hypothesis.