How do you change careers into programming?

An anonymous reader writes: I have worked in tech support for the last several years, but find myself wanting to move on to something else — programming. I've written some small programs in my limited spare time but nothing particularly impressive; just functional stuff to make my life easier. I've spent a lot of time recently working through programming books, and feel I'm ready to make the switch in my career. That said, I don't have a CS degree, and find that responses to my resume have been along the lines of "Thanks, but we aren't hiring for tech support positions." Surely someone from the slashdot crowd has been in the same position — what would you recommend?