My Kyoto Guidebook says:
Flowers bloom throughout the year at Heian Shrine’s beautiful gardens, home of the Jidai Matsuri Festival.
Heian Jingu was built in 1895 to mark the 1,100th anniversary of the founding on Kyoto. It is dedicated to the first, Kammu, and last, Komei, emperors to reign in Kyoto. Heian Jingu is most noted for its wonderful Heian-style gardens, that are arranged around large carp-filled ponds. It is especially well-known for its cherry blossoms in the spring. Shidare Zakura (weeping cherry blossoms) are interspersed with Somei Yoshino Sakura (the most popular variety, nearly white with a touch of the palest pink). See the different varieties of Cherry Blossoms here.
The day I visited Heian Jingu two years ago, it was nice and sunny. Our tour guide explained that we were extremely lucky as the blooms opened that same day, adding that he was worried because the group that came the day before did not enjoy the sakura. The garden paths go around a scenic pond, and there is a very famous bridge along the walk through Shin-en Garden. The park also is also lighted up in the evening but I wasn’t gong to be able to see it I just saw a breathtaking photo of the night scene in a photography book they were selling at the gates.
To get there, take Bus #5 to the Kyoto Kaikan Bijutsu-kan-mae, or take the Tozai Line to the Higashiyama Station. It is a 10-minute walk from the station.