They need both the eyes and the DNA. Eye transplants seem like a relatively safe procedure for ninja, but it's implied that DNA-grafting is a very dangerous process with a high rate or mortality. Moreover, having both clans' DNA only gives one the potential to awaken the Rinnegan.
Obito has had Senju and Uchiha DNA for over a decade without awakening his own Rinnegan, for instance. Nagato, too, was given Madara's eyes in a transplant. He was never given Uchiha DNA, which may be why he couldn't use Uchiha jutsu like Madara does. Thus far, only Madara has truly awoken the Rinnegan like the original Sage.
