We study a bootstrap method which is based on the method of sieves. A linear process is approximated by a sequence of autoregressive processes of order p=p(n), where p(n) tends to infinity, but with a smaller rate than n, as the sample size n increases. For given data, we then estimate such an AR(p(n)) model and generate a bootstrap sample by resampling from the residuals. This sieve bootstrap enjoys a nice nonparametric property. We show its consistency for a class of nonlinear estimators and compare the procedure with the blockwise bootstrap, which has been proposed by Künsch (1989). In particular, the sieve bootstrap variance of the mean is shown to have a better rate of convergence if the dependence between separated values of the underlying process decreases sufficiently fast with growing separation. Finally a simulation study helps illustrating the advantages and disadvantages of the sieve compared to the blockwise bootstrap.