Kodaiji Temple, which is in Higashiyama, is actually pretty easy to miss. I had walked past it numerous times (I remembered the entrance, as you have to walk up quite a bit to go there) but was actually quite surprised when I first went there, as it was really nice. Several of the buildings (I believe) are authentic (as in not rebuilt, but at least one was rebuilt). There are some rock gardens, and you can even get tea served here (there are some famous tea houses here, but they’re historic so you can’t enter them).
Kodaiji temple wasn’t crowded at all, which was also a really nice touch (as places such as Kinkakuji and Kiyomizu are overrun with tourists). There’s a fair bit of walking involved and you have to take off your shoes at points, but this is pretty normal. The entire grounds were really scenic and peaceful, so it was quite nice. Pictures were allowed outside of the buildings, but you couldn’t take pictures in most of them, unfortunately. Also, they had some people explaining parts of the buildings, but these were only in Japanese. There were written explanations as well, but these were also only in Japanese. Even the English language pamphlet is substantially different from the Japanese one (you get both, but the English language one is basically a sheet of paper folded over – the Japanese one is a proper pamphlet).
The entire site takes about an hour to look around and requires a bit of walking (up and down stairs).
Still, Kodaiji Temple is definitely worth it.
Kodaiji temple is open from 9:00 to 5:00 (last entrance, it closes at 5:30). There are no holidays. It costs 600 yen for adults and 250 yen for middle or high school students. Elementary school students are free.
Kodaiji Temple is on a path from Kiyomizu to Yasaka (Gion), so it is rather easy to combine them. It is about a 15 minute walk from Yasaka Shrine, and can be reached from Kyoto Station by bus 206 (get off at Higashiyama Yasui), or can be reached from Kawaramachi by bus 206 (also getting off at Higashiyama Yasui).