Hello Byron,
Everything posted in the replies is right on, I'll add a couple of details.
As mentioned, there are areas beyond Ainkawa such as Harsham and Noble. Prices are around $500-800 per month, unfurnished, 2-3 rooms and 1-2 baths. You will need a car, you can easily get there by taxi but waiting for one to get into the city is a drag. Typical cost to and from Harsham or Noble is around $4-$6.
Bakhtiary is great, big houses (3 floors, 4+ rooms), supermarkets, restaurants, and coffee houses. Nothing as nice as Ainkawa as far as restaurants go, and definitely no alcohol. Prices start at $800 and go as high as $2500 per month or more (for really big houses). You shouldn't have any problems finding anything within your budget of $1000 per month. Also, they're building a movie theater and luxury mall there.