No, you can definitely fly with non-service dogs. I've flown with my Jack Russell terrier in a dog carrier in the cabin with me a ton of times, but not so much since 9/11. Back then, all you had to do was buy a pet ticket when you bought your own ticket ($50 or something?), and get a letter/form from the vet saying your dog is good to travel. You have to let the airline know in advance because sometimes they only allow 2 in-cabin pets per leg of the flight.
These were the rules pre-9/11; i would check with individual airlines about their policies before booking flights. For example,
here is Delta's webpage about traveling with pets in-cabin.