"Black holes and time warps: Einstein's outrageous legacy" mentions a version of GR that's equivalent to standard GR (in the sense that it makes the same predictions about experiments), but describes spacetime as flat. In this theory, matter doesn't curve spacetime, it deforms measuring devices.
I also think that one of the early attempts at a quantum (field) theory of gravity could accurately reproduce the predictions of GR. (Split the metric into a flat part plus the deviation from flatness, g=η+h, quantize only h, and throw away all Feynman diagrams that contain loops).
'Field Theory Gravity' models gravity as a tensor field in Minkowski space. There's a good discussion by Y. Baryshev in arXiv:gr-qc/9912003 v1 (1 Dec 1999). It is generally believed that FTG is identical to GR but Baryshev disputes this and claims they make different predictions in the high-energy regime.