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Spectral methods have proven invaluable in numerical simulation of partial differential equations but the frequent global communication required raises a fundamental barrier to their use on highly parallel architectures. To explore this issue, we implemented a three dimensional implicit spectral method on an Intel hypercube. Utilization of about 50% was achieved on a 32 node iPSC/ 860 hypercube, for a 64 x 64 x 64 Fourier spectral grid; finer grids yield higher utilizations. Chebyshev-spectral grids are more problematic, since plane- relaxation based multigrid is required. However, by using a semicoarsening multigrid algorithm, and by relaxing all multigrid levels concurrently, relatively high utilizations were also achieved in this harder case. In fact, since the amount of work per processor was higher in this case, we achieved somewhat higher utilization, typically 60% on moderate sized problems. Thus spectral methods remain attractive on the current generation of distributed memory architectures.