I did a lot of thinking on this topic back in my post, "Of Radagast" or "Radagast the Brown" or whatever I named it. "Success of the Wizards", that's what it was called.
I've always, ALWAYS been under the assumption that Radagast wasn't exactly idle over there in Wilderland, and almost certainly (living so near) played a large role in the rise of the Beornings, keeping the Misty Mountains goblin-free, spying on Dol Goldur (also something he was quite close to), communicating with the elven wardens on the borders of Lothlorien, maintaining the levels of spiders in the forest and preventing them from spreading as much as possible, helping the woodmen, lakemen, Dale-men, and dwarves of the mountain as much as possible.
It's pretty clear that he kept his ears open and relayed a lot of information back and forth via his friendship with all the eagles and other birds (ravens, thrushes), thereby preparing the whole of Wilderland for war (which did indeed come to them).
I think he was probably asked to join the White Council, even if he declined, but as such, I think he completed his task as far as combatting Sauron.
It seems like each wizard sort of picked an area, or group to assist in their cunning, wizardly ways, and stuck with them using the best means to do so depending on each area of the map.
Gandalf went big name, acting as counsellor to all Numenor related lands (Gondor, Arnor) and was associated frequently with elves of all variety, but none more than the Deep Elves, lords like Elrond.
The other two wizards went east.
Radagast set up shop in Wilderland.
Anyway, it seems clear that being a sort of 'cousin' to Radagast was how Gandalf thought he could earn the alliance of Beorn - a chief among Wilderland's people. Which would mean, that Radagast's name precedes him in those lands as a reliable counsellor and ally, much like Gandalf's did all over his territories (of course, like Eomer mentioned, it was no longer the key to Theoden's favor).
I covered it VERY extensively in my Radagast topic, which can be found here;
http://forum.barrowdowns.com/showthread.php?t=2094