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 K\"{u}nsch (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.