For the pressure (or free energy) of QCD, four-dimensional (4d) lattice data is available at zero baryon density up to a few times the critical temperature T-c. Perturbation theory, on the other hand, has serious convergence problems even at very high temperatures. In a combined analytical and three-dimensional (3d) lattice method, we show that it is possible to compute the QCD pressure from about 2T(c) to infinity. The numerical accuracy is good enough to resolve in principle, e.g., logarithmic contributions related to 4-loop perturbation theory.