The concepts of fractal tiling and hyperbolic tiling are combined to create novel fractal surfaces in Euclidean three-space. Paper folding and computer modeling are used to create these constructs. We show examples using triangular, trapezoidal, and dart-shaped prototiles. Smaller tiles deflect out of the plane of adjacent larger tiles, resulting in non-planar surfaces, the shapes of which are dependent on the rules governing the sign of the deflections as well as their magnitudes. These constructs are an intriguing blend of organic and geometric character and in some cases bear marked resemblance to natural leaves.