We propose a novel scheme for trapping ultracold rubidium and ytterbium atoms in a three-dimensional (3D) optical lattice simultaneously, in which the two species of atoms locate on two staggered lattices with the same spatial period and have a spatial separation of 133 nm. Furthermore, we calculate the tunneling and intra- and interspecies interactions of rubidium and ytterbium atoms as a function of light intensity, and find that the mixture of quantum degenerate gases in optical lattices can exhibit more intriguing quantum phases, especially a staggered dual Mott insulator of alkali-metal and alkaline-earth metal atoms.