Recent progress in the synthesis of π-conjugated porphyrin arrays of different shapes and dimensionalities motivates us to examine the band structures of infinite (periodic) porphyrin nanostructures. We use screened hybrid density functional theory simulations and Wannier function interpolation to obtain accurate band structures of linear chains, 2D nanosheets, and nanotubes made of zinc porphyrins. Porphyrin units are connected by butadiyne (C4) or ethyne (C2) linkers or “fused” (C0), i.e., with no linker. The electronic properties exhibit strong variations with the number of linking carbon atoms (C0/C2/C4). For example, all C0 nanostructures exhibit gapless or metallic band structures, whereas band gaps open for the C2 or C4 structures. The reciprocal space point at which the gaps are observed also show fluctuations with the length of the linkers. We discuss the evolution of the electronic structure of finite porphyrin tubes made of a few stacked six-porphyrin rings toward the behavior of the infinite nanotube. Our results suggest approaches for engineering porphyrin-based nanostructures to achieve target electronic properties.