A simple method is presented for the rapid simulation of statistically isotropic non-Gaussian maps of cosmic microwave background temperature fluctuations with a given power spectrum and analytically calculable bispectrum and higher-order polyspectra. The nth-order correlators of the pixel values may also be calculated analytically. The cumulants of the simulated map may be used to obtain an expression for the probability density function of the pixel temperatures. The statistical properties of the simulated map are determined by the univariate non-Gaussian distribution, from which pixel values are drawn independently in the first stage of the simulation process. We illustrate the method using a non-Gaussian distribution derived from the wavefunctions of the harmonic oscillator. The basic simulation method is easily extended to produce non-Gaussian maps with a given power spectrum and diagonal bispectrum.