Guest houses for foreigners, commonly called "gaijin houses" are convenient as you don't need a guarantor, a long-term visa or to pay the normal 4 months deposit and key money. They can be rented monthly or sometimes even on a weekly or daily basis. They are usually good places to meet people.
More numerous than guest houses. They usually require a guarantor + deposit and/or key money. The advantadge is being able to choose any type, size and location you want. The following links tend to be more foreigner oriented and also have short-term and guarantor-free apartments.
Short & long stay apartments and guesthouses at various locations mainly in West-Central Tokyo.
Nord House Real estate
Apartment search engine in English for Tokyo's 23 wards. Some short-term apartments as well.
Kimi will be
Daily, weekly, monthly apartments and guest houses.
J and F Accommodation
Plenty of guesthouses, apartments and monthly mansions, mainly in West-Central Tokyo.
Upmarket Serviced Apartments
Bureau
Comfortable serviced apartment in central Tokyo. Site and service in English and Japanese.
Weekly and Monthly Mansions (Self-catering Apartments)
Weekly and monthly mansions are used mainly by Japanese people. They are often more expensive and not always better than "gaijin houses". They enjoy the same advantages (no deposit, no guarantor, no furniture needed and can be rented for the short term), but if you want to meet people, that is definitely not the place to be. The attraction is that they are much more numerous than "gaijin houses", which makes it possible to choose almost any neighbourhood in Tokyo. However, you will need to read and speak some Japanese to find them. The following sites have the most extensive and user-friendly listing. Just choose your area, price range, and desired size.