Coronal hard X-ray (HXR) and continuum ensuremathγ-ray sources associated with the impulsive phase of solar flares have been the subject of renewed interest in recent years. They have been interpreted in terms of thin-target, non-thermal bremsstrahlung emission. This interpretation has led to rather extreme physical requirements in some cases. For example, in one case, essentially all of the electrons in the source must be accelerated to non-thermal energies to account for the coronal HXR source. In other cases, the extremely hard photon spectra of the coronal continuum ensuremathγ-ray emission suggest that the low-energy cutoff of the electron energy distribution lies in the MeV energy range. Here, we consider the role of inverse Compton scattering (ICS) as an alternate emission mechanism in both the ultra- and mildly relativistic regimes. It is known that relativistic electrons are produced during powerful flares; these are capable of upscattering soft photospheric photons to HXR and ensuremathγ-ray energies. Previously overlooked is the fact that mildly relativistic electrons, generally produced in much greater numbers in flares of all sizes, can upscatter extreme- ultraviolet/soft X-ray photons to HXR energies. We also explore ICS on anisotropic electron distributions and show that the resulting emission can be significantly enhanced over an isotropic electron distribution for favorable viewing geometries. We briefly review results from bremsstrahlung emission and reconsider circumstances under which non-thermal bremsstrahlung or ICS would be favored. Finally, we consider a selection of coronal HXR and ensuremathγ-ray events and find that in some cases the ICS is a viable alternative emission mechanism.