I should add that Funny Games is an intensely disturbing film. It frequently gets mentioned whenever theres’s a “What’s the scariest/most disturbing film you’ve ever seen?” thread.
Oh yeah, one other thing: it’s in German.
The 1967 movie, Wait Until Dark, is exceedingly creepy, and downright terrifying at the end. (And all without any blood).
The Devil’s Backbone is atmospheric, creepy, scary, gut-wrenching, and with almost no gore.
Not to be confused with Devil’s Rain or Devil’s Rejects or Devil’s Frying Pan, or any of the other Devils.
I second haardvark’s recommendation for The Changeling. I saw it years ago, when it first came out. Watched it again a few months ago and it was just as scary, it held up just fine.
The original Haunting (1963) based on Shirley Jackson’s book is also very scary, with no blood.
The Others (Nicole Kidman) might be along your lines. What’d you think of Sixth Sense?
The Shining
I suggest reading the book instead.
You have some very difficult criteria. You don’t like “seeing people being tortured, dismembered or struck with blunt (or sharp) weapons”, yet you like the violent Alien, which includes the title creature exploding out from a man’s chest, killing him.
Including some that others have mentioned:
Rosemary’s Baby, The Innocents, The Others, Poltergeist, The Legend of Boggy Creek, Close Encounters of the Third Kind, Wait Until Dark, Night of the Demon, Duel, The Exorcist, Race with the Devil, The Collector.
Check out Strange Days (1995) which is a thriller with some very tense, scary moments. Be warned, depending on your sensibilities…
…there’s a very graphic and disturbing rape scene.
There’s the original Nosferatu. It’s not really scary, but it is quite creepy. It feels like you’re watching a real vampire.
Then you can see Shadow of the Vampire, a fictional movie about the making of Nosferatu where the vampire is played by a real vampire.
Shadow of the Vampire not ‘scary’ so much as disturbing. One of the best horror films I’ve seen in a long time. However it’s all about mood so it has a long long build up.
True, it’s not really horror. I found it closer to dark comedy. But it’s worth seeing if you’ve seen Nosferatu.
It’s worth seeing if you haven’t as well. Great movie just one you should have an idea of what you’re going into.
I’ve never liked excessive violence/gore either, but I still like a good scare every once in awhile.
I second Wicker Man – good production values despite a low budget, and almost no violence. Beware of the cheesy '70s sexploitation scenes though. 
The original Night of the Living Dead is squeaky clean by today’s standards, and since it was filmed in grainy black and white, what little gore you see is relatively easy to stomach. Very suspenseful.
I don’t remember the original Nightmare on Elmstreet being too graphic, although it’s been awhile. I know it scared the hell out of me though.
For movies that are more creepy than terrifying, I nominate Donnie Darko and Pi.
How about the part where the girl kills her parent with the garden trowl, then eats the heart?
For Creepy and little gore, some suggestions-
Clownhouse
The Believers
Hideaway (Pretty good Dean Koontz book, not so great movie.)
John Carpenter’s original 1980 version of The Fog is good. A little bit of gore, but not too much. I also enjoy the weirdness of Carpenter’s In the Mouth of Madness and Prince of Darkness.
These are all pretty old though, and have been suggested before, but they’re all classics in their own rights.
I’ll third (fourth?) “Rosemary’s Baby” – great movie; no gore. “The Tenant” is another great Polanski movie in that category, though IIRC it does get a bit more explicitly bloody. And when I think about it, Polanski’s does a lot of this kind of movie: you could also count “Repulsion,” “Knife in the Water,” and even “Chinatown” on the list.
If you want to count suspense movies among the scary/terrifying, you’ll have lots to choose from – ie, basically all of Alfred Hitchcock’s films, to begin with.
If you don’t mind “Alien” level violence, you should have no problem with “Silence of the Lambs.” There’s not actually that much gore in it.
One of the best creepy, atmospheric horror movies I’ve seen in a long time is The Woman in Black.
Okay, well “squeaky clean” was probably a bad choice of words. But I wouldn’t put the gore level much higher than Alien, and certainly nowhere near Romero’s later films.
I third this, and for exactly the same reasons.
I tend to refer to this film as “asylum porn” instead of horror. Lots of neat scenery in it. The movie itself isn’t half bad either. 
Haven’t seen it, and I never will (the fact it was writen and directed by a convicted child molester, who molested an actor in the movie will keep me away thanks), but I also laughed at the description: “Just before Halloween, three young brothers alone in a big house are menaced by three escaped mental patients who have murdered some traveling circus clowns and taken their identities.” Can there be anything more trite, tired, and stupid?
Let’s see:
Halloween… check.
Alone teenagers… check.
Big, scary house… check.
Escaped mental patients… check.
Murders… check.
Clowns… check.
Sounds more horrible, than horror.