We report the structures and physical properties of two new iridates, NaIrO3 and Sr3CaIr2O9, both of which contain continuous two-dimensional honeycomb connectivity. NaIrO3 is produced by room temperature oxidative deintercalation of sodium from Na2IrO3, and contains edge-sharing IrO6 octahedra that form a planar honeycomb lattice. Sr3CaIr2O9, produced via conventional solid-state synthesis, hosts a buckled honeycomb lattice with novel corner-sharing connectivity between IrO6 octahedra. Both of these new compounds are comprised of Ir(5+) (5d(4)) and exhibit negligible magnetic susceptibility. They are thus platforms to investigate the origin of the nonmagnetic behavior exhibited by Ir(5+) oxides, and provide the first examples of a J = 0 state on a honeycomb lattice.