E*Trade offers no-fee ATM withdrawals from any ATM. They also have some kind of rate on their Max Checking account. The drawback is that unless you live near one of their 10 branches in Northeast, they have no physical presence.
I believe Citizens Bank has some rewards when you use their checking and debit card.
But I recommend finding a credit union. I have a small credit union with 4.15% APY on my checking with no minimums and rebates on ATM fees, etc. Some other credit unions also offer rewards points with their debit card...
my USAA debit card has rewards. AFAIK the best debit rewards are even more marginal than the best cc rewards.
I have the Wells Fargo rewards program attached to my debit card, but I don't think it's all that impressive. $12 per year to participate, and you receive 1 point for every $4 you spend. You have to actually sign for the transaction to receive points. If you use your PIN for the purchase, you don't receive points.
It ends up being something like.. spend $2600, get $5 cash reward or $5 gift certificate.