  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% < 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 ~ (1+z) is
also plausible and could result from selection effects. From the observed
luminosity-temperature (L-T) relation, L ~ T_obs^(2.82+-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 ~ 0.5--4.
