Love suspense so here you go:
Another - I personally had some issues with the show but it is enjoyable and provides both suspense and horror elements and keeps you guessing. Threads were full of different theories at the time of airing.
Aoi Bungaku Series are a set of mini stories. They are not really horror but all of them have a hidden idea behind that makes you think. It's mostly drama though some stories do have horror elements.
Death Note, that's a given. Will be surprised if you haven't seen it already. It is all about mind games and keeps you guessing as to what will happen next. Main chara finds a notebook and whoever s name he writes in will die.
Elfen Lied - gory stuff with limbs getting cut off. As well as sad with some mind tricks.
Ga-Rei Zero is not horror but it is a bloody show with some interesting topics. Besides the way things are handled is very different from what I've seen in most action series. Really worth a watch.
Ghost Hunt - is one of my faves. You learn a lot of new stuff about ghost hunting. Series are separated in different arcs/cases. Some better than others. Labyrinth arc is my fave one. Though second arc will keep you at the edge as well
Highschool of the Dead - ZOMBIES! That alone says it all. One minus, too much fanservice.
Hundred Stories - has a very unique and different animation style what makes it feel much creepier than it would with normal style. Series are about Japanese legends and demons. Some episodes can be disturbing.
Jigoku Shoujo - are episodic series with mini story later on. Theme is very similar to Death Note, is it fine to kill bad people? Basically there is this girl who takes revenge for other people.
Mahou Shoujo Madoka★Magica - not a horror but psychological and full of twists and suspense. Cute art is very deceiving. Watched series in one go. Happens rarely for me.
Monster - not horros, detective series so suspense is there. A surgeon had once saved boy's life and years later learn that the boy has become into a serial killer. Surgeon feels responsible and tries to find him besides his accused of being a killer so he has to run and hide.
Phantom: Requiem for the Phantom - not a horror but suspense, drama, great soundtracks, interesting characters and great animation are included
I personally adore this series.
Shiki - this series are slightly different. You as a viewer know who is the culprit but not the characters so you are stuck watching how village is fighting for their survival, every characters takes their own approach. Suspense included. Watched it one go too.
Steins;Gate - surprised that no one mentioned it considering it's popularity. Series are about time travelling and how tiny little changes in the past can change everything. Not a horror but second half will keep you at the edge of your seat.
Umineko no Naku Koro ni - horror with bunch of disturbing killings and suspense.
That's all I have for now, hope it helps.