  We present three-dimensional, self-consistent radiative transfer solutions
with a new Monte Carlo radiative equilibrium code. The code, RADISHE
($\bf{RAD}$iative transfer $\bf{I}$n $\bf{S}$moothed particle
$\bf{H}$ydrodynamics and $\bf{E}$ulerian codes), can be applied to calculate
the emergent spectral energy distributions (SEDs) and broadband images from
optical to millimeter wavelengths of arbitrary density geometries with
distributed sources of radiation. One of the primary uses of this code has been
to interface with hydrodynamical codes to calculate emergent SEDs along a
simulation time sequence. The primary methodological focus of this paper is on
the radiative equilibrium temperature calculation. We find that an iterative
calculation of the temperature, which takes as the Monte Carlo estimator for
the mean free intensity the sum of photon flight paths, is significantly faster
than relaxation temperature calculation methods, particularly when large
numbers of grid cells are required, i.e., in modeling three-dimensional
geometries such as the dust envelopes of turbulent massive protostellar cores
or infrared bright galaxies. We present simulated color-color plots for
infrared bright galaxies at a range of redshifts, and unfold these plots as
color vs the fractional AGN luminosity, to demonstrate that $\it{Herschel}$
will be able to effectively discriminate between submillimeter galaxies where
the energy source is dominated by AGN and those where star formation dominates.
[abridged]
