Frequency Filtered Random Noise : this technique uses a Fourier transform to generate random periodic textures. See also image deconvolution.

(* runtime: 20 seconds *)
n = 275; SeedRandom[1]; ListDensityPlot[Abs[InverseFourier[Fourier[Table[Random[], {n}, {n}]] Table[Exp[-((j/n - 0.5)^2 + (i/n - 0.5)^2)/0.025^2], {i, 1, n}, {j, 1, n}]]], Mesh -> False, Frame -> False, ColorFunction -> (Hue[# + 0.5] &)];
Here is some Mathematica code for an animated version. This technique can be used for simulating water surfaces:
(* runtime: 18 seconds *)
n = 64; SeedRandom[0];
Scan[ListDensityPlot[#, Mesh -> False, Frame -> False, ColorFunction -> (Hue[# + 0.5] &)] &, Abs[InverseFourier[Fourier[Table[Random[], {32}, {n}, {n}]]Table[Exp[-((i/n - 0.5)^2 + (j/n - 0.5)^2 + (t/32 - 0.5)^2)/0.025^2], {t, 1, 32}, {i, 1, n}, {j, 1, n}]]]];