The aim of this paper is to deal with Poisson noise in images arising in electron microscopy. We consider here especially images featuring sharp edges and many relatively large smooth regions together with smaller strongly anisotropic structures. To deal with the denoising task, we propose a variational method combining a data fidelity term that takes into account the Poisson noise model with an anisotropic regulariser in the spirit of anisotropic diffusion. In order to explore the flexibility of the variational approach also an extension using an additional total variation regulariser is studied. The arising optimisation problems can be tackled by efficient recent algorithms. Our experimental results confirm the high quality obtained by our approach.