Most of the big sign-up bonuses come from various credit cards. It is possible to get some miles through checking accounts (I just got 25,000 CO miles by opening a Chase checking account last month). I also opened a CO credit card at the same time and expect to get another 25,000 miles in sign-up bonuses through that.

I'm earning some miles through my debit card (1 mile for every $2), but I'm earning them twice as fast on my credit card. Plus, some credit cards offer an even higher earning ratio.

In short, yes, you can earn miles without a credit card by taking advantage of debit cards and other offers. It's just a little faster to rack up the miles with a credit card. I use a credit card myself quite frequently, but I pay it off in full each month to avoid interest charges.