Successfully tracking the human body is an important perceptual challenge for robots that must work around people. Existing methods fall into two broad categories: geometric tracking and direct pose estimation using machine learning. While recent work has shown direct estimation techniques can be quite powerful, geometric tracking methods using point clouds can provide a very high level of 3D accuracy which is necessary for many robotic applications. However these approaches can have difficulty in clutter when large portions of the subject are occluded. To overcome this limitation, we propose a solution based on fully convolutional neural networks (FCN). We develop an optimized Fast-FCN network architecture for our application which allows us to filter observed point clouds and improve tracking accuracy while maintaining interactive frame rates. We also show that this model can be trained with a limited number of examples and almost no manual labelling by using an existing geometric tracker and data augmentation to automatically generate segmentation maps. We demonstrate the accuracy of our full system by comparing it against an existing geometric tracker, and show significant improvement in these challenging scenarios.