We have used the Submillimeter High Angular Resolution Camera (SHARC) bolometer camera at the Caltech Submillimeter Observatory to map the distribution of the broadband 350 mu m continuum emission toward the Orion A molecular cloud. A comparison of the 350 mu m flux densities in OMC-1 with previous 1100 mu m measurements indicates a strong spatial variation of the grain emissivity exponent, beta. The lowest value of the exponent (beta similar or equal to 1.75) is found toward the Orion Bar photon-dominated region (PDR), while the highest value (beta similar or equal to 2.5) is found toward the Orion Ridge north of IRc2. This variation is consistent with the destruction of grain mantles by the UV photons from the Trapezium cluster. The observed spatial variation of beta in OMC-1 suggests that the long-wavelength grain emissivity may also vary significantly in GMC cores on small linear scales (less than or similar to 0.5 pc), affecting H-2 column density and mass estimates. The 350 mu m continuum emission in the Orion Bar region correlates well with the CO (6-5) peak brightness temperature and is shifted by similar to 10 " from the molecular component traced by the (CO)-C-13 (6-5) emission. This indicates that the 350 mu m dust emission in this region originates predominantly in the outer high-temperature PDR layers. Several filamentary structures previously detected in molecular tracers are also seen in our map at flux levels comparable to those seen in the Bar. Over 30 compact dust sources are detected in the OMC-2 and OMC-3 clouds, including a dozen sources not previously known. The average 350/1300 mu m flux ratio based on our data and previous observations of this region (63 +/- 19) indicates low dust temperatures (17 +/- 4 K, assuming beta = 2) for most of the sources. The brightest 350 mu m source in OMC-3 has a low 350/1300 mu m ratio (similar to 23), indicating a very low dust temperature (similar to 10 K), or a significant opacity at 350 mu m (similar to 2.5). This source appears to be a deeply embedded and cold young protostar. A comparison of the mass estimate for the OMC-2/3 filament based on the 350 mu m continuum emission with previous (CO)-O-18 mass estimates indicates a relatively high grain emissivity, Q(350) = 4 x 10(-4), in this region.