In Granada, I stayed at Hostal Landazuri (sp?) on Cuesta de Gomerez. It is very close to the Alhambra (300m, all uphill), and close to the bus stop for the Alhambra-Albaycin route. They have a choice between in-room and down the hall bath and toilet facilites, so you can be a little more flexible with price.
The rooms are basic and clean and staff is very nice and speaks English. The central heat is kept on low though, and I did find the room a bit cold (I was there in January, probably not an issue now). There is a little cafe downstairs that serves a reasonably priced breakfast, but is only open until 10:30 am.
Address: Cuesta Gomérez, 24
Phone 958 22 14 06
You should also check out previous posts on these cities. I know we had a Granada thread going in February...
Hope that helps!