Hi Paul, Many probably don't realize that by "rewards checking", you're referring not just to interest-bearing checking accounts, but a specific product type that pays VERY high interest in return for specific beahvior. To answer your question, no. Credit Unions and smaller banks only offer rewards checking. One reason for this is that BancVue, the company who helps design these products, targets smaller institutions looking to improve their loyalty and grow their asset base.
I'm guessing one reason you might like a larger bank would be to insure banking with these products is convenient for you. One workaround: pick a credit union that accepts deposits via ATMs and partakes in the "shared branching network". This will allow you to do most anything at any credit union that belongs to these groups (and most credit unions do). This is how I effortlessly use a few different rewards checking accounts with institutions outside my region. If you're interested, I wrote a more in depth review of rewards checking accounts on my blog.