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Exact marginal inference in continuous graphical models is computationally challenging outside of a few special cases. Existing work on approximate inference has focused on approximately computing the messages as part of the loopy belief propagation algorithm either via sampling methods or moment matching relaxations. In this work, we present an alternative family of approximations that, instead of approximating the messages, approximates the beliefs in the continuous Bethe free energy using mixture distributions. We show that these types of approximations can be combined with numerical quadrature to yield algorithms with both theoretical guarantees on the quality of the approximation and significantly better practical performance in a variety of applications that are challenging for current state-of-the-art methods.