Fourier phase retrieval is a classical problem of restoring a signal only from the measured magnitude of its Fourier transform. Although Fienup-type algorithms, which use prior knowledge in both spatial and Fourier domains, have been widely used in practice, they can often stall in local minima. Modern methods such as PhaseLift and PhaseCut may offer performance guarantees with the help of convex relaxation. However, these algorithms are usually computationally intensive for practical use. To address this problem, we propose a novel, unsupervised, feed-forward neural network for Fourier phase retrieval which enables immediate high quality reconstruction. Unlike the existing deep learning approaches that use a neural network as a regularization term or an end-to-end blackbox model for supervised training, our algorithm is a feed-forward neural network implementation of PhaseCut algorithm in an unsupervised learning framework. Our network is composed of two generators: one for the phase estimation using PhaseCut loss, followed by another generator for image reconstruction, all of which are trained simultaneously using a cycleGAN framework without matched data. The link to the classical Fienup-type algorithms and the recent symmetry-breaking learning approach is also revealed. Extensive experiments demonstrate that the proposed method outperforms existing approaches in Fourier phase retrieval problems.