In an approach similar to Ewald’s method for evaluating lattice sums, we split the application of Helmholtz Green’s functions between the spatial and the Fourier domains and, for any finite accuracy, approximate their kernels. In the spatial domain we use a near optimal linear combination of decaying Gaussians with positive coefficients and, in the Fourier domain, a multiplication by a band-limited kernel obtained by using new quadratures appropriate for the singularity in the Fourier domain. Applying this approach to the free space and the quasi-periodic Green’s functions, as well as those with Dirichlet, Neumann or mixed boundary conditions on simple domains, we obtain fast algorithms in dimensions two and three for computing volumetric integrals involving these Green's functions. This is a joint work with Chris Kurcz amd Lucas Monzon.