How to delete messages off your iPhone?

I have an iPhone 3GS and I delete All the conversations but the messages stop show a fragment when I search words that were in the messages. Help! How can I dispose of the messages completely?

Added (1). Deleted* still*
Freaking auto correct.

Added (2). Gene! I did that but they still show up when I search them

Don't know

Go to Message's. Edit. And click the red line next to all of the messages and it will clear the conversation you had with that person and delete them off of your iPhone.