A novel phase sequence for the transition from the double diamond to the double gyroid cubic phases via two non-cubic intermediate phases, an orthorhombic Fmmm (O69) phase and a hexagonal P63/m (H176) phase, is reported for specifically designed bolapolyphiles composed of a linear rod-like bistolane core with sticky glycerol ends and two branched central and two linear peripheral side chains. These liquid crystalline (LC) phases represent members of a new class of unicontinuous network phases, formed by longitudinal rod bundles with polar spheres acting as junctions and the alkyl chains forming the continuum around them. In contrast to previously known bicontinuous cubic networks, they combine different junctions with different angles in a common structure, and one of them even represents a triple network instead of the usually found double networks. This provides new perspectives for the design of soft network phases with enhanced structural complexity, inspiring the search for new supramolecular networks, nano-particle arrays, and photonic band-gap materials.