My personal favorite is the Hostal Van Gogh. It is on Miguel de Ma~nara, just steps away the Real Alcazar and a hop, skip and a jump from the Cathedral. I actually got the last room there two years ago for Semana Santa, and to my surprise, it was right on the route of some of the processions.
Only drawback is that if you get the room on the ground floor with the bathroom in the hall, the bathroom is too small. I had to stand on the closed toilet to get dressed. Get a room on the other floors or the one with the bathroom on the ground floor.
The group also has other hostels, which I will assume are of the same quality. Great and cheap. http://www.grupo-piramide.com/