  We propose a family of free fermion lattice models that have "Dirac loops",
closed lines of Dirac nodes in momentum space, on which the density of states
vanishes linearly with energy. Those lattices all possess the planar trigonal
connectivity present in graphene, but are three dimensional. We show that their
highly anisotropic and multiply-connected Fermi surface leads to quantized Hall
conductivities in three dimensions for magnetic fields with toroidal geometry.
In the presence of spin-orbit coupling, we show that those structures have
topological surface states. We discuss the feasibility of realizing the
structures as new allotropes of carbon.
