I got a non-technical post-graduate degree and now I'm a programmer. Only took a couple of years to get my first programming job. Here's how I did it.
First of all, I did as much programming as I could at my tech support jobs. Not all of it was company sponsored, but if I figured out something I could write that would help *me* do my job I would write it. I wrote all kinds of little things, and then I was able to truthfully add to my resume that I developed software.
I was also going to user group meetings for the language I was using most and meeting people there. I ended up getting my first job (and all subsequent jobs, actually) through people I met at those meetings. At least for the language, city, and time I happened to be in, the meetings were filled with people who knew about more work than they could take. And the recommendations you can get there are worth "2-5 years of experience" on a resume.
I'm currently helping my company's QA guy get some programming tasks so he can make the switch and give his job to some other poor CS grad. Is there anyone on the development team where you are that might help you out?
There's one more option: recruiters. I know they're not great, and the jobs you get through them aren't all perfect, but there are some recruiters who can help you market yourself without the exact "2-5 years of experience" someone's looking for.
One last thing: If you're any good at all you'll be way ahead of most people in this field. If you can get an interview, showing your abilities and desire to learn can be enough.