The last we heard, he told Gandalf that Saruman wanted to talk to the Grey Wizard. Then he sent Gwaihir to find Gandalf. Then he just disappeared, and was never heard from again in the books.
After the Council of Elrond, scouts from Rivendell journey to Radagast's home of Rhosbobel near the eaves of southern Mirkwood and find it abandoned. There is no record of what became of the Brown Wizard after he sent the Eagle Gwaihir to Isengard. Saruman might have had him murdered. Or, after learning of Saruman's treachery, Radagast might have gone into self-exile. No one really knows, though it seems unlikely that he returned to Valinor unless it was many years later.