Supernova remnants (SNRs) are widely believed to be the principal source of Galactic cosmic rays, produced by diffusive shock acceleration in the environs of the remnant's expanding blast wave. Such energetic particles can produce gamma rays and lower energy photons via interactions with the ambient plasma. The recently reported observation of TeV gamma rays from SN 1006 by the Collaboration of Australia and Nippon for a Gamma-Ray Observatory in the Outback (CANGAROO), combined with the fact that several unidentified EGRET sources have been associated with known radio/optical/X-ray-emitting remnants, provides powerful motivation for studying gamma-ray emission from SNRs. In this paper, we present results from a Monte Carlo simulation of nonlinear shock structure and acceleration coupled with photon emission in shelllike SNRs. These nonlinearities are a by-product of the dynamical influence of the accelerated cosmic rays on the shocked plasma and result in distributions of cosmic rays that deviate from pure power laws. Such deviations are crucial to acceleration efficiency considerations and impact photon intensities and spectral shapes at all energies, producing GeV/TeV intensity ratios that are quite different from test particle predictions. The Sedov scaling solution for SNR expansions is used to estimate important shock parameters for input into the Monte Carlo simulation. We calculate ion (proton and helium) and electron distributions that spawn neutral pion decay, bremsstrahlung, inverse Compton, and synchrotron emission, yielding complete photon spectra from radio frequencies to gamma-ray energies. The cessation of acceleration caused by the spatial and temporal limitations of the expanding SNR shell in moderately dense interstellar regions can yield spectral cutoffs in the TeV energy range that are consistent with Whipple's TeV upper limits on those EGRET unidentified sources that have SNR associations. Supernova remnants in lower density environments generate higher energy cosmic rays that produce predominantly inverse Compton emission observable at super-TeV energies, consistent with the SN 1006 detection. In general, sources in such low-density regions will be gamma-ray-dim at GeV energies.