We propose and investigate a realization of the position- and momentum-correlated Einstein-Podolsky-Rosen (EPR) states [Phys. Rev. 47, 777 (1935)] that have hitherto eluded detection. The realization involves atom pairs that are confined to adjacent sites of two mutually shifted optical lattices and are entangled via laser-induced dipole-dipole interactions. The EPR "paradox" with translational variables is then modified by lattice-diffraction effects, and can be verified to a high degree of accuracy in this scheme.