The noise that lavalier microphones produce when rubbing against clothing (typically referred to as rustle) can be extremely difficult to automatically remove because it is highly non-stationary and overlaps with speech in both time and frequency. Recent breakthroughs in deep neural networks have led to novel techniques for separating speech from non-stationary background noise. In this paper, we apply neural network speech separation techniques to remove rustle noise, and quantitatively compare multiple deep network architectures and input spectral resolutions. We find the best performance using bidirectional recurrent networks and spectral resolution of around 20 Hz. Furthermore, we propose an ambience preservation post-processing step to minimize potential gating artifacts during pauses in speech.