This is an anthology of three stories. The first two, Yotsuya Kaidan and Tenshi Monogatari, each consisting of 4 episodes, are both adaptations, though I can't vouch for whether or not they are faithful. The final segment, Bakeneko, at only three episodes, appears to be an original story, though just as with the other two, it's influenced by Japanese folklore and pre-20th century Japanese art periods.
Although there might be some important distinctions between the terms, Ayakashi is similar to the word Yokai, which is a type of demon or spirit. Like most anime listed as horror, these are probably not stories a person into western, or even eastern, horror would seek out. Yes, they have demons, ghosts, violence and gore, and are rather grotesque, but they're really more discernible as supernatural tragedies.
To begin with, the opening for the series features the strange choice of a rap song with Japanese-styled background music. The Samurai Champloo creators were able to make the hip-hop tracks fit the show, despite the music style and time period being anachronistic. Here, it's just odd and the song doesn't mesh well at all. The ending theme and other music tracks are much more appropriate and fit the darker tone of the episodes.
Yotsuya Kaidan is a tragedy with very reprehensible characters. Most of them are highly immoral and cruel. Probably the only remotely sympathetic characters are the child and the two exploited women. This particular story bears somewhat of a resemblance to Masaki Kobayashi's Kwaidan anthology, with the rejected wife of the callous and ambitious samurai dying and becoming a vengeful spirit.
However, it manages to become fairly convoluted with its two intersecting stories and multiple plot twists. There are some loose threads, such as the grownup child appearing in one scene and never appearing again. But worst of all is that none of the characters are compelling, and whether they are wicked or sympathetic, they're all equally flat. As it is unlikely for most viewers to become invested in the plight of the characters, the tragedy that permeates the piece remains hollow, if not completely artificial.
The animation is generally as good as it needs to be for such a story, though it obviously never rises to the level of Hakkenden or Akira. The production design is striking in many scenes, most notably the dusk scenes where the male protagonist leaves everything behind. The contrast of the saturated red sky and the silhouetted surroundings, with some selective illumination upon a solitary grave, is very ominous. Lightning is also used to excellent effect. All of the scenes at the house surrounded by graves are also visually striking.
As enjoyable as I found the adaptation to be at times, it all feels rather pointless by the end. A very standard revenge story once you peel away all of the aesthetic flourishes. It seems the writer felt the same way—that the story didn't hold up on its own—for he employed metanarrative devices by the end. Each episode has a narrator, who enters into the story very directly after what should be the end sequence. Alongside live action footage, occasionally with bits of animation, he meditates on the nature of the story being told. Essentially, the ghost is to the characters as the story is to those who experience it. Yet this element doesn't mesh very well and seems more like a half-baked idea than anything profound.
Tenshi Monogatari is a tale of a human and a demon-like woman (referred to as a "forgotten god") falling in love. The woman is the head of a female clan responsible for devouring humans. The man is attempting to climb the ranks of society and also has a wife (or mistress). Both characters have something to lose that provides conflict throughout, and there is an exploration of the demonic becoming more human and vice versa.
It's a predictable story every step of the way, featuring saccharine music for the romantic parts, along with a happy end that had me snorting milk. The production design is rather solid, especially in the first episode. It's clear that by the end there's too much going on to be animated effectively.
The highlight is the third segment, a mystery about a botched wedding of two affluent families and a medicine seller who possesses magical means to vanquish a cat spirit that begins haunting the family, but before he can unlock his powers, he has to solve the mystery behind the spirit. This spawned a series called Mononoke the next year, featuring the medicine seller (as well as the maid, Kayo, in one arc) and the same distinct style, reminiscent of a luridly colored Ukiyo-e painting in motion. Similar to Gankutsuou, the animation is never distractingly bad, but it's the artwork that carries it. The story isn't anything terribly original, but it's a well-presented mystery. The pacing is also much tighter than the other stories because of its shorter length, though it may leave the piece feeling slightly rushed.
For fans of Mononoke or anyone who is intrigued by the art style and presentation of Bakeneko, the preceding segments can be skipped, as they're all independent stories. The previous two are worth a look as well, but if you've seen similar Japanese ghost stories, they may seem a bit humdrum.