In Season 3, Episode 9, "Rains of Castamere", Bran asks Rickon and Osha to head to the Last Hearth, seat of House Umber, who are sworn to House Stark. They left in the night, separating from Bran, Hodor, Meera and Jojen.
Note that Jon Umber "the Greatjon", head of House Umber, unlike in the books, is not present at the Twins during the events of the Red Wedding, making him one of the few loyal bannermen of House Stark that remains alive and free. This makes Last Hearth a safe place for Rickon to stay in.
This is the related dialogue among Bran, Rickon and Osha in S03E09, "Rains of Castamere" (emphasis mine):
Bran: (to Osha) I'm not asking you to come with me. It won't be safe for Rickon.
Rickon: Me? I'm coming with you.
Bran: No. You and Osha and Shaggydog head for the Last Hearth.
Bran: The Umbers are our bannermen. They'll protect you.
Rickon: I'm coming with you. I'm your brother. I have to protect you.
Bran: Right now I have to protect you. Robb's at war and I'm going beyond the Wall. If something happens to us, you're the heir to Winterfell.
Bran: (to Osha) Would you know how to find the Last Hearth?
Osha: You southerners build your big castles and you never move. You're easy to find.
In Season 6, Episode 3, "Oathbreaker" it has been revealed that: Rickon and Osha made it to the Umbers, and that Jon Umber "the Greatjon" is dead. The current head of House Umber (Smalljon Umber) is not loyal to House Stark and has delivered Rickon and Osha to Ramsay Bolton in Winterfell.
Rickon and Osha made it to the Umbers, and that Jon Umber "the Greatjon" is dead. The current head of House Umber (Smalljon Umber) is not loyal to House Stark and has delivered Rickon and Osha to Ramsay Bolton in Winterfell.
