I've now lived 5-1/2 months without an alarm clock. We lost it somewhere between here and Croatia, and we never bothered to replace it.
It has a bit of an effect on my workday. Sometimes I get in at 7:30am, sometimes at 9:30, sometimes 8:30. But I like it. It's liberating. I just get up when my body tells me to get up.
I was planning on replacing it until I read Jeff Davidson's advice in The Complete Idiot's Guide to Managing Your Time (don't laugh, it's a good book): "Give up your alarm clock!... Any time it wakes you up, you don't truly know how long you would have slept."
So I have a good handle of my sleep needs. 8-1/2 to 9 hours a night. Get rid of your alarm clock. Try it, you'll like it. You might end up looking as well rested as me.