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Most research on deep learning algorithms for image denoising has focused on signal-independent additive noise. Focused ion beam (FIB) microscopy with direct secondary electron detection has an unusual Neyman Type A (compound Poisson) measurement model, and sample damage poses fundamental challenges in obtaining training data. Model-based estimation is difficult and ineffective because of the nonconvexity of the negative log likelihood. In this paper, we develop deep learning-based denoising methods for FIB micrographs using synthetic training data generated from natural images. To the best of our knowledge, this is the first attempt in the literature to solve this problem with deep learning. Our results show that the proposed methods slightly outperform a total variation-regularized model-based method that requires time-resolved measurements that are not conventionally available. Improvements over methods using conventional measurements and less accurate noise modeling are dramatic - around 10 dB in peak signal-to-noise ratio.