We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian(x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule–width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)-coordinate finite differencing, while those in the y≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.