Many implicit schemes have been proposed for direct numerical simulation of particles in fluids, but they can be quite complicated and require a lot of memory when the concentration of particles is high. Explicit finite difference schemes on a regular grid are one way around these issues that has not yet been proposed. The fluid velocity and density can be marched in time, and the particles can be moved according to Newton's laws. Compressible flows can be exactly simulated, and the scheme is easily parallelized. Enforcing the no-slip condition on the spherical particle's surfaces as they move on a Cartesian grid is not so easy, however, so we have proposed a spectral expansion method that exactly satisfies no-slip to deal with the problem, improving on prior methods that required matrix inversions and iteration. This method allows a grid as coarse as ten grid spacings per particle diameter to give smooth forces and accurately resolved pressure distributions on the particle surface. The method has been validated by comparison to a finite element particle solver, and by direct comparison to experiment. Up to 1035 particles have been simulated on a PC with four cores, and the effective viscosity of a sheared particulate suspension has been calculated up to a particle concentration of 25%.