Hello Bern
Here are a few ideas:
Very cheap, clean and new: K’s House Kyoto
Backpackers Hostel K’s House Kyoto
418 Nayacho, Shichijo-agaru, Dotemachi-dori, Shimogyo-ku,
Kyoto city, Japan 600-8142
TEL +(81)-75-342-2444 FAX +(81)-75-342-2440
email: kyoto@kshouse.jp
http://kshouse.jp/index_e.html
Pretty Cheap, friendly, clean: The Three Sister’s Inn & The Three Sister’s Inn ANNEX
Near Heian Shrine in a quite neighborhood convenient for bus and subway about 15 minutes on foot.
tel: +(81)-75-761-6333 fax +(81)-75-761-6335
email: kyo-3sis-anx@air.ocn.ne.jp
(no website it seems)
Reasonably Priced: Kyoto Weekly / Monthly Mansion and Apartment
Our friends have stayed here and it is nice, reasonably priced and clean. It looks like their site is only available in Japanese. (These are just apartments with no ‘Kyoto’ character.)
http://www.monthly-kyoto.jp/
Higher End: Iori Kyoto Machiya Stays (traditional Kyoto townhouse stays)
Iori Kyoto Machiya is great, not cheap, but a very special and spacious experience. Plenty of charm!
http://www.kyoto-machiya.com/eng/