We introduce a regularization method that gives a smooth formulation for the fundamental solution to Stokes flow driven by an infinite, triply-periodic array of point forces. With this formulation, the velocity at any spatial location may be calculated, including at and very near the point forces; these locations typically lead to numerical difficulties due to the singularity within the Stokeslet when using other methods. For computational efficiency, we build upon previous methods in which the periodic Stokeslet is split into two rapidly decaying sums, one in physical space and one in reciprocal, or Fourier, space. We present two validation studies of our method. First, we compute the drag coefficient for periodic arrays of spheres with a variety of concentrations of sphere packings; and second, we prescribe a force density onto a periodic array of spheres, compute the resulting nearby velocity field, and compare these velocities to those computed using an immersed boundary method formulation. The drag coefficients computed with our method are within 0.63 % of previously published values. The velocity field comparison shows a relative error of about 0.18 % in the L2-norm. We then apply our numerical method to a periodic arrangement of sinusoidal swimmers. By systematically varying their spacing in three directions, we are able to explore how their spacing affects their collective swimming speed. © 2012 Elsevier Inc..