The fabrication flexibility of 3D printing has sparked a lot of interest in designing structures with spatially graded material properties. In this paper, we propose a new type of density graded structure that is particularly designed for 3D printing systems based on filament extrusion. In order to ensure high-quality fabrication results, extrusion-based 3D printing requires not only that the structures are self-supporting, but also that extrusion toolpaths are continuous and free of self-overlap. The structure proposed in this paper, called CrossFill, complies with these requirements. In particular, CrossFill is a self-supporting foam structure, for which each layer is fabricated by a single, continuous and overlap-free path of material extrusion. Our method for generating CrossFill is based on a space-filling surface that employs spatially varying subdivision levels. Dithering of the subdivision levels is performed to accurately reproduce a prescribed density distribution. We demonstrate the effectiveness of CrossFill on a number of experimental tests and applications.