Hi, welcome to AZ!
For Cuban, I haven't been, but I hear that Havana Cafe is good. A bit pricey compared to what I'm used to from the East Coast. http://www.havanacafe-az.com/
For Greek, we like My Big Fat Greek Restaurant, a few locations. I don't consider restaurants with more than one location a chain, but you might. Great appetizers and you can even get out cheaply by getting the sandwich platters instead of dinner platters. http://www.azeats.com/mybigfatgreek/
Great place for Subs in general is Jersey Mikes. They're a chain, but they slice the cold cuts to order, have authentic jersey-style subs, and their cheesesteak is good too. http://www.jerseymikes.com/
For quick Mexican, this may not be in your area, but Los Taquitos was on Diner's Drive-Ins and Dives, and is excellent and cheap. They're near the I-10 on Elliott Rd., toward Chandler. Definitely worth checking out at least once. True family mom-and-pop place.