Using a new lattice polymer simulation method for block copolymers, we have studied a dense polymer system composed of monodisperse linear A-B-C triblock copolymers. We confirm that a tricontinuous structure for linear A-B-C block copolymers observed by Mogi et al. (Macromolecule 25, 5408 (1992) and ibid. 25, 5412 (1992)) could be, indeed, an ordered tricontinuous “double-diamond” (OTDD) structure. In addition, two (symmetric and shifted) OTDD phases and a lamellar phase have been observed in the same simulations. The two OTDD phases are conceivable by shifting one diamond network against the other diamond network.