Although we have yet to determine whether the DAMA data represents a true discovery of new physics, among such interpretations inelastic dark matter (IDM) can match the energy spectrum of DAMA very well while not contradicting the results of other direct detection searches. In this paper we investigate the general properties that a viable IDM candidate must have and search for simple models that realize these properties in natural ways. We begin by determining the regions of IDM parameter space

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... that are allowed by direct detection searches including DAMA, paying special attention to larger IDM masses. We observe that an inelastic dark matter candidate with electroweak interactions can naturally satisfy observational constraints while simultaneously yielding the correct thermal relic abundance. We comment on several other proposed dark matter explanations for the DAMA signal and demonstrate that one of the proposed alternatives -- elastic scattering of dark matter off electrons -- is strongly disfavored when the modulated and unmodulated DAMA spectral data are taken into account. We then outline the general essential features of IDM models in which inelastic scattering off nuclei is mediated by the exchange of a massive gauge boson, and construct natural models in the context of a warped extra dimension and supersymmetry.