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In this paper, we propose an interlaced multi-shell sampling scheme for the reconstruction of the diffusion propagator from diffusion weighted magnetic resonance imaging (DW-MRI). In standard multi-shell sampling schemes, sample points are uniformly distributed on several spherical shells in q-space. The distribution of sample points is the same for all shells, and is determined by the vertices of a selected polyhedron. We propose a more efficient interlaced scheme where sample points are different on alternating shells and are determined by the vertices of a pair of dual polyhedra. Since it samples more directions than the standard scheme, this method offers increased angular discrimination. Another contribution of this work is the application of optimal sampling lattices to q-space data acquisition and the proposal of a model-free reconstruction algorithm, which uses the lattice dependent sinc interpolation function. It is shown that under this reconstruction framework, the body centered cubic (BCC) lattice provides increased accuracy. The sampling scheme and the reconstruction algorithms were evaluated on simulated data as well as rat brain data collected on a 600 MHz (14.1T) Bruker imaging spectrometer.