Wichita National Wildlife Refuge -- we were able to take our pooch rock-hopping and hiking on most of the trails. No mention was made of leashes, though discerning people wouldn't let Fido near the prairie dogs or buffalo (they bite, kick, and could otherwise damage Fido).
Nordaggio's Coffee is a great little cafe located in south Tulsa, near Oral Roberts University. I bring my dog with me there frequently, to lie at my feet while I work. The staff is really great about it. The cafe is frequented by a lot of college students, who love seeing (and petting!) a dog in there. I've been told the owner is a dog-lover. You can sit outside at tables, or inside the cafe itself.
Pei-Wei restaurant at Kingspointe Village shopping center at 61st and Yale in Tulsa OK has an outdoor patio where you can eat with your pets after you've ordered your food inside (at least they didn't ask me to leave with mine and I don't need the T-shirt, I'm grateful to have found your site)