Demons - That movie scared the bejebus outta me when I was a kid. I still get freaked watching it now. I don't think I'd watch it alone.
Alien - I agree with RD. This movie was/is creepy.
Cube - It was a small Canadian production done by some film students out of Toronto. Don't let that turn you off though; it is very creepy. Root Down or Freewheelin' may have seen it. There were some sequels, but none were as well done as the original. It actually stars Ezri Dax from Deep Space Nine. Super cutie.

Aside from that, I would have to mention The Exorcist, 28 Days Later...I'm drawing a blank here. I'll have to think some more.
Oh and you're not the only one Cash who didn't think The Ring was very scary. The only part that freaked me out was that one brief camera shot of the girl in the closet who saw the ghost. Eek. The face was gruesome.
Anyways, I love horror movies. I'm always longing for good sci-fi and horror movies, or sci-fi/horror (i.e. Alien and Sunshine).
I'll have to put Sunshine in the "strange movie" thread.