What phone are you using? There is a way to do so with Verizon Messages Integrated Messaging, but I never used that method. Start with the shortest thread and delete the conversation. You can choose more than one at a time, but this is going to take a LONG TIME. Your phone will freeze up and stop responding. I recommend you do so just before bedtime. Select all and then delete. By morning you should be clear. Keep plugged into charger.
You can back up message threads you want to keep to Verizon Cloud, or another cloud service you prefer.
I've deleted larger threads and let my phone sit to complete the process which certainly works well as Snn5 has said.
I've also gone in and selected the largest thread, and then long pressed a message and had to delete them in large groups of 30. I had to do this for 15 minutes and then finally was able to delete the message as a whole. It's definitely more tedious.
I was also under the impression that deleting the data under the app might delete some threads, I have threads I need to keep at the moment so haven't tested the theory.