We present 350 micron observations of 15 Chapman et al. submillimeter galaxies (SMGs) with radio counterparts and optical redshifts. We detect 12 and obtain sensitive upper limits for three, providing direct, precise measurements of their far-infrared luminosities and characteristic dust temperatures. With these, we verify the linear radio--far-infrared correlation at redshifts of z ~ 1--3 and luminosities of 10^11--10^13 L_sun, with a power-law index of 1.02+-0.12 and rms scatter of 0.12 dex. However, either the correlation constant q or the dust emissivity index beta is lower than measured locally. The best fitting q ~ 2.14 is consistent with SMGs being predominantly starbust galaxies, without significant AGN contribution, at far-infrared wavelengths. Gas-to-dust mass ratios are estimated at 54+-13 (kappa_850 / 0.15 m^2 kg^-1), depending on the absoption efficiency kappa, with intrinsic dispersion ~40% around the mean value. Dust temperatures consistent with 34.6+-3 K (beta/1.5)^-0.71, at z ~ 1.5--3.5, suggest that far-infrared photometric redshifts may be viable, and perhaps accurate to 10%