We analyze the magnetoresistance ratio of finite and infinite iridium and platinum chains. Our calculations, which are based on a combination of the nonequilibrium Green's function techniques and density-functional theory, include a fully self-consistent treatment of noncollinear magnetism and of the spin-orbit interaction. They indicate that, in addition to having an extremely large magnetic anisotropy, infinite and also realistic finite-length iridium chains show sizeable anisotropic magnetoresistance ratios. We therefore propose iridium nanostructures as promising candidates for nanospintronics logic devices.