I did a bit of research today and learned that this happens to a lot of people! Apparently when your estrogen levels drop right before your period, the muscles that surround your urethra temporarily experience a drop in pressure (you can read about this here: http://kidney.niddk.nih.gov/kudiseases/pubs/uiwomen/).
When I have insurance again, this is something I'll get checked out as it's kind of a pain in the butt and it might be a sign of other problems. (I don't have diabetes or any kind of infection, though, I did get that seen to before my insurance ran out.) Just wanted to share the info here in case other people experienced the same thing.
FWIW, I'm 35 and nulliparous. I can remember this being an issue for me for roughly the last ten years.