In this paper, we introduce TBARF (Turning BAnd Random Fields), a RF generation algorithm based on the turning band method that is optimized for massively parallel hardware such as GPUs. Our algorithm replaces the 3D FFT with a lower order, one-dimensional FFT followed by a projection step, and is further optimized with loop unrolling and blocking. We show that TBARF can easily generate RF on non-regular (non uniform) meshes and can afford mesh sizes bigger than the available GPU memory by using a streaming, out-of-core approach. TBARF is 2 to 5 times faster than the traditional methods when generating RFs with more than 16M cells. It can also generate RF on non-regular meshes, and has been successfully applied to two real case scenarios: planetary nebulae and cosmological simulations.