We apply the optical nanocircuit concepts to design and analyze in detail a three-dimensional (3D) plasmonic nanotransmission line network that may act as a relatively broadband negative-refraction metamaterial at infrared and optical frequencies. After discussing the heuristic concepts in our theory, we show full-wave analytical results of the expected behavior of such materials, which show increased bandwidth and relative robustness to losses. The possibility and constraints of getting a 3D fully isotropic response are also explored and conditions for minimal losses and increased bandwidth are discussed. Full-wave analytical results for some design examples employing realistic plasmonic materials at infrared and optical frequencies are also presented, and the case of a subwavelength imaging system using a slab of this material is numerically investigated.