The search for alias-free sampling lattices for a given frequency support, in particular those lattices achieving minimum sampling densities, is a fundamental issue in various applications of signal and image processing. In this paper, we propose an efficient computational procedure to find all alias-free integral sampling lattices for a given frequency support with minimum sampling density. Central to this algorithm is a novel condition linking the alias-free sampling with the Fourier transform of the indicator function defined on the frequency support. We study the computation of these Fourier transforms based on the divergence theorem, and propose a simple closed-form formula for a fairly general class of support regions consisting of arbitrary N-dimensional polytopes, with polygons in 2-D and polyhedra in 3-D as special cases. The proposed algorithm can be useful in a variety of applications involving the design of efficient acquisition schemes for multidimensional bandlimited signals.