I discuss ongoing work that aims to provide combinatorial tools that give explicit free resolutions of normalizations of toric ideals. We use stratifications and a modified version of the resolutions of lattice ideals introduced by Bayer and Sturmfels to produce resolutions. These techniques construct resolutions analogous to those in the work of Hanlon, Hicks, and Lazarev. As a consequence, we show that the resolutions of the diagonal for toric varieties given by Hanlon, Hicks, and Lazarev agree with those given by Brown, and Erman.