From my experience the best way to find cheap property to rent in Gran Canaria is to speak Spanish and look in La Provincia or a similar local newspaper. Tips & Info and Round Town News are free papers too and you can find these in bars, shops and restaurants.
Another option is to use one of the property agencies in Gran Canaria but then you will have to pay the equivalent of half a month or a whole month’s rent in fees. Of course if you find the perfect house or apartment in Gran Canaria it can be worth doing that if you are planning to stay a long time (or if your Spanish isn’t up to par!)
The best way to find somewhere to live is to come over with a package holiday then at least you have somewhere for a couple of weeks while you look. In Gran Canaria property to rent tends to be widely available and you can find a very small 1 bedroom apartment for 400 euros (absolute cheapest) or something a bit bigger. You normally have to pay 1 month’s rent and 1 month’s deposit to move in, although this does vary.