They only accommodate guests with certain social status but they are more generous for American tourists. And just as what's happening to first class western style hotels, the price of these premium Japanese inns has also gone down from the way I knew before. Tourist Information at JR Kyoto Station has a list of such Inns. Here are English speaking ones that are popular among foreign travelers. They are not as classy or authentically 'Japanese' as they may appear to you.
Some temples accommodate tourists and offers interesting Buddhist experience.
You may or may not be asked to attend their Buddhist morning activities such as cleaning and meditation depending on their policy. Here is the rare English speaking one in an excellent location with fascinating historical background for an exceptionally inexpensive price. I stayed there myself and found it comfortable, too. Click to see photos and more details. Some may have some private rooms, but most are 'dormitory' in which you will have to share a room with 2-14 other people.