For a highly recommended taste of Japanese cuisine, try the Karyo restaurant in Gion near Yasaka Jinka. The menu is completely in Japanese, but order the evening kaiseki course meal when you place your reservations and prepare for a memorable dining experience. There is meticulous attention to every detail, from the elegant entrance to the dining areas. Meals change daily depending upon the freshest ingredients available, and reflect the season. Reservations are required, but your hotel concierge should be able to arrange this.
Plan on spending several hours savoring the balance of flavors and the beautiful presentation in Karyo. The 10-course menu costs ¥10500 per person, and contains all of the traditional elements of a true kaiseki dinner. The website is only in Japanese, but there are good photographs, an access map, prices, and reservation information: http://www.karyo-kyoto.jp/index.html