Hi, Alexx --
It depends what you mean by cheap. In February, I stayed at the Hostal Cruz Sol
in the Plaza Santa Cruz, near the Plaza Mayor and the Puerta del Sol. I loved it and would stay there again. The place was clean and well maintained, and the staff was pleasant. The woman at the desk spoke excellent English. A room with an ensuite bathroom cost me less than 300 euros for the week, and that included fees for twenty or so phone calls to the States. I forget whether they had cheaper rooms without the ensuite toilet.
I find the best resource when I'm looking for affordable accommodations in Spain is the Spain/Portugal volume of the LET\'S GO
books. Just make sure you get the most recent edition.