While Gadgetfreak and others have brilliant suggestions for dining places, I would like to add a few general observations from time spent in the aforementioned cities.
Most Japanese consider Kyoto to be far from the coast and do not really encourage to try Sushi there. Of course Kyoto has many Sushi bars of every price category but it's not what this ancient capital is renowned for. Another Kyoto-related issue is the rather distanced approach born and bred Kyoto residents show even towards Japanese from other regions. The dialect spoken in Kyoto is quick, witty and presents an ideal tool for gossip and chatter.
The Kaiseki cuisine is one local speciality that consists of numerous small portions of seasonal tidbits and visual beauty. While it alone is reason enough to book a Ryokan in Kamigyo-ku or Northern Kyoto, many Kaiseki restaurants remain elitist bastions that make Parisian three star gourmet temples look like easy going family chains. If accompanied or introduced by a well-connected local host they can be an enjoyable experience but otherwise it might be fairly difficult to even book a table at some establishments. So I would second Gadgetfreak and recommend booking a well known Ryokan for the whole experience including a Kaiseki dinner.
In summertime the mountains North of Kyoto host Kaiseki restaurants with tatami mat terraces placed in the shadow of trees and sometimes over mountain streams. These al fresco eateries are a wonderful opportunity to escape the summer heat of the Kyoto valley.
Other Kyoto specialities include (white and brown) Miso, various pickles and several thriving French (or French-Japanese fusion) restaurants.
There is even a splendid semi-Parisian cáfe almost opposite from the Kyoto Hotel serving perfect continental petit dejeuner.
There are countless restaurants on the historical alleys adjoining the Kamo river on the Kamigyo-ku side, some are Kaiseki legends with Geishas and hundreds of years of history but for the non-Japanese visitor the Izakayas ( gastropubs of Japan
) are of more interest. Some have beautiful gardens preserved from Imperial times but almost all present innovative Japanese dishes and a great selection of Rice Wine and Shochu ( the heavy stuff). Gion on the other side has become more of a cluster of bars and nightclubs in the very broad sense.
And please don't forget the restaurants on the top floors of major department stores. Kyotoites themselves use them a lot, so they can't really be that bad. Also, the food sections of prime department stores like Takashimaya are very pleasant places for foodies. Another place to visit is the indoor food market hidden in the shopping alleys between Shijo and Sanjo streets. I even managed to find the ingredients for a ratatouille complete with French herbs there once.
For more budget-oriented eating the surroundings of the Doshisha and Kyoto University campuses just north of the Imperial Palace offer many places geared towards student tastes and wallets. Some are run by really fascinating people involved in human rights groups etc. so if you want to find a little Berkeley touch in Japan this is the place for it.
Ps. Mosburger has MANY outlets in Kyoto
