We present 350 \mu m 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 \sim 1 –3 and luminosities of 10 ^ { 11 } – 10 ^ { 13 } L _ { \odot } , with a power-law index of 1.02 \pm 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 \simeq 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 ^ { +14 } _ { -11 } \ > \bigl ( \kappa _ { 850 \mu { m } } / 0.15 m ^ { 2 } kg ^ { -1 } \bigr ) , depending on the absoption efficiency \kappa _ { \nu } , with intrinsic dispersion \simeq 40 \% around the mean value . Dust temperatures consistent with 34.6 \pm 3 K \ > ( \beta / 1.5 ) ^ { -0.71 } , at z \sim 1.5 –3.5 , suggest that far-infrared photometric redshifts may be viable , and perhaps accurate to 10 \% \lesssim dz / ( 1 + z ) , for up to 80 \% of the SMG population in this range , if the above temperature characterizes the full range of SMGs . However , observed temperature evolution of T _ { d } \propto ( 1 + z ) is also plausible and could result from selection effects . From the observed luminosity-temperature ( L - T ) relation , L \propto T _ { obs } ^ { 2.82 \pm 0.29 } , we derive scaling relations for dust mass versus dust temperature , and we identify expressions to inter-relate the observed quantities . These suggest that measurements at a single wavelength , in the far-infrared , submillimeter or radio wave bands , might constrain dust temperatures and far-infrared luminosities for most SMGs with redshifts at z \sim 0.5 –4 .