You can have what you want Infraction Jack. There are options to have a smartphone without a data plan.
Page Plus is a good option, they have the standard plan in which you use your cash balance to make calls or texts and any calls come out of the balance at whatever the rate you are setup on (the more you spend at a time to refill your account, the cheaper it is per minute) You do not need to use texts on this plan and you will not pay anything if you don't, but it is there if you need it. Same with 3g internet, albeit at a high price. The one thing that AstroManLuca forgot to mention is that 4g phones are not compatible with Page Plus yet, except in rare cases where someone manages to successfully reprogram the phone for it to work.
My other recommendation is Airvoice Wireless, which uses the AT&T network. They have a $10 per month plan that can be used any way you want, such as all voice calls only, or some combination of voice, texts, and data. Their website details the rate structure. Again, you do not have to use or pay for any texting here if you don't want to. Any AT&T smartphone can be used, as far as I know. Not sure about LTE phones but phones such as the iphone and the Galaxy s2 are fine. Your best bet is to go on ebay and pick up a sim card for a buck and then all you have to do is call airvoice and activate it.
Since you never stated a budget for a phone, I suggest some sort of iphone on Airvoice. Or some Android smartphone. Page Plus is more limited in options since the latest phones for Verizon are 4g so it is slim pickins for 3g only phones that are any good compared to what is out now.