AT&T and several other carriers have prepaid (Android) smartphones. I have one and simply didn't buy a data plan. So it doesn't work at all unless I am in a hotspot or similar free area and then I can turn the data on manually.
What many people don't realize is that it's not the carrier at fault here but Apple. Apple mandates in their contracts with the providers that every iPhone has to have a data plan included. So if you go to a different type/brand of smartphone, you're able to order services individually.
AT&T's prepaid service is actually quite decent. I bought an Android phone for $99 and that was that. I can go to the app store, buy programs, use the GPS, the FM radio (it's not web-based radio), and have all of the functions of a smartphone other than data. For not a dime per month. Essentially I have a $99 iPod Touch since the apps are identical in almost every case.
Verizon doesn't have prepaid smartphones. None of the 10 cent a minute options are real smartphones. If you get a tablet or smartphone, they require a data plan.
T-Mobile does let you do this. But their phones are much more money, typically.
NOTE - you usually must buy a phone that is prepaid. Many can be upgraded to full service but almost none of the contracted/full service phones can be downgraded to prepaid. This isn't a technological problem but a contractual one with the phone makers and the providers. Yes, the prepaid can be downgraded and upgraded each month as you need them to be, if that's what you desire.