We report millimeter and submillimeter continuum observations of six FU Orionis stars (FUor) and one possible pre-FUor star. Based on five-point maps at 800-mu-m, we find resolved sizes of 37" x 12" for V346 Nor and 18" x 24" for Elias 1-12. Results from aperture photometry at 800-mu-m for Elias 1-12 and at 800 and 850-mu-m for Z CMa are also consistent with resolved sources. We combine the measured submillimeter to millimeter continuum with mid-infrared and far-infrared observations to model the source of thermal emission in these sources. Our disk and spherical cloud models include power-law forms for the radial density (SIGMA is-proportional-to r-p) and temperature (T is-proportional-to r-q) profiles and for the frequency dependent profile of the emissivity (Q is-proportional-to nu-beta) at long wavelengths. Knowledge of q and of the submillimeter spectra allow us to place constraints on the possible values that the beta index can take on. In every case, the spherical cloud model fails because the implied visual optical depths are greater than 10(2) along the line of sight to the star, and all of these stars are optical sources with [V] < 20. The long-wavelength spectra from all seven stars-FU Ori, Z CMa, V346 Nor, V1515 Cyg, V1331 Cyg, V1057 Cyg, and Elias 1-12-are best modeled as arising from optically thin disks containing between 2 x 10(-2) and 0.3 M. of gas plus dust. The emission from V1331 Cyg is optically thick out to lambda > 2 mm, hence the true mass of its disk could be larger. We find minimum disk radii of r(disk) > 35 AU for FU Ori and 1700 AU > r(disk) > 45 AU for the other FUors. In most cases, the disk is optically thick if viewed edge-on at 800-mu-m. Therefore, our viewing angle must be closer to pole-on for most of these FUors, and this may in fact provide evidence for previous assertions that FUors are preferentially detected based on these favorable viewing angles. We find that the distribution of the power-law indices for the temperature profiles is very different from that found for T Tauri stars. We also find that in three cases the emissivity index may be less than unity, although in all cases solutions with beta greater-than-or-equal-to 1 may be found, and for three FUors only beta greater-than-or-equal-to 1.5 is reasonable. Our results suggest that the disks around FUors are active, not passive, and that the material in the disks may be accretionally unevolved. Together, these conclusions indicate that FUors may be very young T Tauri stars.