A stable volume integral equation formulation based on equivalent volumetric currents is presented for modeling electromagnetic scattering of highly inhomogeneous dielectric objects. The proposed formulation is numerically solved by means of Galerkin method of moments on uniform grids, allowing for acceleration of the matrix–vector products associated with the iterative solver with the help of FFT. In addition, the pertinent volume–volume Galerkin inner products are reduced to purely surface–surface integrals with smoother kernels, allowing for highly accurate and fast computation by readily available sophisticated cubatures. Numerical results demonstrate the convergence properties of the algorithm for scatterers with high dielectric contrast. In addition, we describe a road- map for Magnetic Resonance-specific volume integral equations fast solvers based on the proposed algorithm.