A new type of carbon nanotube, based on the graphenylene motif, is investigated using density functional and tight-binding methods. Analogous to conventional graphene-based nanotubes, a two-dimensional graphenylene sheet can be "rolled" into a seamless cylinder in armchair, zigzag, or chiral orientations. The resulting nanotube can be described using the familiar (n,m) nomenclature and possesses 4-, 6-, and 12-membered rings, with three distinct bond lengths, indicating a nonuniform distribution of the electron density. The dodecagonal rings form pores, 3.3 Å in diameter in graphenylene, which become saddle-shaped paraboloids in smaller-diameter nanotubes. Density functional theory predicts zigzag nanotubes to be small-band gap semiconductors, with a generally decreasing band gap as the diameter increases. Interestingly, the calculations predict metallic characteristics for armchair nanotubes with small diameters (<2 nm), and small-band gap semiconducting characteristics for larger-diameter ones. Graphenylene nanotubes with indices mod(n-m,3) = 0 exhibit a band gap approximately equal to that of armchair graphenylene nanotubes with comparable diameter.