A new method for the numerical simulation of three-dimensional incompressible flows is described. Our vortex-in-cell (VIC) method traces the motion of the vortex filaments in the velocity field which these filaments create. The velocity field is not calculated by creating a mesh-record of the vorticity field, then integrating a Poisson's equation via the fast Fourier transform to get the stream function and generating a mesh-record of the velocity field. The computed scales of motion are assumed to be essentially inviscid. Viscous or subgrid-scale effects are incorporated into a filtering procedure in wave vector space. Three computational experiments were pursued in three-dimensional space. The velocity of translation of a single vortex ring was measured and compared with the Biot-Savart law. Next, the evolution in time of an infinite periodic array of closed vortex filaments (Taylor-Green) was studied. The third simulation follows a mixing layer from an initial state of uniform vorticity with two- and three-dimensional small perturbations. Streamwise perturbations lead to the usual roll-up of vortex patterns with spanwise uniformity.