Craigslist has been a godsend for apartment hunting here in Minneapolis, I bet it'd work well in the Ft Worth area too. My gf's neighbors in her duplex have a big-headed intimidating-looking bully breed with huge teeth and a big goofy grin that shows off those pearly whites.
He'd likely scare anyone who isn't familiar with dogs, or at least anyone who isn't an animal lover.. but to me he's just a big goofball, he just loves people and would never intentionally hurt anyone. It made me happy to know that the landlord is a guy who meets-and-greets potential renters AND their dogs before giving approval. I got to talk to him, he said that he's more than happy to rent to dog owners, and doesn't care about breeds, but will not allow openly aggressive dogs.
Keep looking, I'm sure there are places out there and they probably aren't even that hard to find, if you're willing to compromise on other aspects of housing (price, size, location, how "nice" it is, etc) then it should be even easier.
If you can find duplexes/fourplexes with something for rent, or something that is run by a single owner as a small bit of extra income, you might have more luck than looking at big apartment compelxes.