  We present a generic algorithm for generating Gaussian random initial
conditions for cosmological simulations on periodic rectangular lattices. We
show that imposing periodic boundary conditions on the real-space correlator
and choosing initial conditions by convolving a white noise random field
results in a significantly smaller error than the traditional procedure of
using the power spectrum. This convolution picture produces exact correlation
functions out to separations of L/2, where L is the box size, which is the
maximum theoretically allowed. This method also produces tophat sphere
fluctuations which are exact at radii $ R \le L/4 $. It is equivalent to
windowing the power spectrum with the simulation volume before discretizing,
thus bypassing sparse sampling problems. The mean density perturbation in the
volume is no longer constrained to be zero, allowing one to assemble a large
simulation using a series of smaller ones. This is especially important for
simulations of Lyman-$\alpha$ systems where small boxes with steep power
spectra are routinely used.
  We also present an extension of this procedure which generates exact initial
conditions for hierarchical grids at negligible cost.
