Given a correlation function c(x) with x in Rn, the turning bands method (TBM) for n=2 and 3 constructs realizations of an n-dimensional and stationary process Y~N(0, c(x)) from appropriately summed line processes. Therefore an implementation of TBM calls naturally for fast and accurate generations of line realizations. These have generally been generated by a spectral approach since the Fourier transform of the line covariances is linked in a very simple fashion to the N-dimensional Fourier transform of c(x). However, we show that in many instances a space domain implementation of TBM, with line realizations obtained from a circulant embedding method, has advantages over a spectral implementation in that without increase in computational costs (1) the line realizations display exactly the required covariance structure, (2) knowledge of the spectral density of the process Y is not required, and (3) fine tuning of line process parameters is not needed. ¿ American Geophysical Union 1995