Laplacian

I wish there were a tool that would quickly solve Laplacians on a regular two-dimensional grid, for example in the NetPBM format.

Under what conditions can the Laplacian be solved "directly" using the Fourier transform, and under what must some sort of numerical method (e.g., multigrid) be used? Is is necessarily a computational "win" to use the Fourier tranform?

UPDATE: my solver using relaxation (or Successive Over-Relaxation)