We present exact analytical results for properties of anomalous diffusion on a fractal mesh. The fractal mesh structure is a direct product of two fractal sets, one belonging to a main branch of backbones, the other to the side branches of fingers. Both fractal sets are constructed on the entire (infinite) y and x axes. We suggest a special algorithm in order to construct such sets out of standard Cantor sets embedded in the unit interval. The transport properties of the fractal mesh are studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation 〈x^{2}(t)〉≃t^{β}, where the transport exponent β<1 is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with β>1 has been observed as well when the environment is controlled by means of a memory kernel.