I would actually be hard pressed to name more than a few true mountains that require ropes to get to the summit. Nearly every mountain in AZ has non-technical scrambling routes to the top.
Maybe you are referring the peaks that have established trails to the top that at times get pretty steep? Off trail scrambling can be pretty brutal (cactus, catclaw, etc).
Just to pitch in with suggestions that are almost in season: Lots of people seem to enjoy Mazatzal Peak, and you might like Peeley and Sheep as well, as well as perhaps North Peak and others near the Mazzy Divide trail.
Hike Arizona it is full of sharp, pointy, ankle-twisting, HAZmaster crushing
Hike Arizona it is full of sharp, pointy, shin-stabbing, skin-shredding
Hike Arizona it is full of striking, biting, stabbing, venomous