The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inverse transform method, when combined with a Hilbert transform approach for computing the cdf of the Lévy increment, is reliable and efficient. The Hilbert transform representation for the cdf is easy to implement and highly accurate, with approximation errors decaying exponentially. The inverse transform method can be combined with quasi-Monte Carlo methods and variance reduction techniques to greatly increase the efficiency of the scheme. As an illustration, discrete Asian options pricing in the CGMY model is considered, where the combination of the Hilbert transform inversion of characteristic functions, quasi-Monte Carlo methods and the control variate technique proves to be very efficient.