Places to play: Hibiya Park (Tokyo Metro Courts)
 
Hibiya Koen (park) has 5 hard courts, all with lights.  Despite being next door to the office buildings of Kasumigaseki, Yurakucho and Marunouchi, there's lots of greenery around and the atmosphere is quite pleasant.  The courts are close enough to many foreigners' offices that it's possible to play tennis during a (bit lengthy) lunch period, as Chris Alderson and I have done a few times.  The changing rooms and shower are rather minimal and grubby, though. 

To get use, see the info page on the Tokyo Metro lottery system.  Remember you can check for untaken times quite separately from the monthly lottery itself.

The admin office is right next to the courts.  Go here before going on the courts.

Hibiya Koen is at the southeast corner of the Imperial Palace, across the street from the Imperial Hotel.  Just walk around the park to find the courts.  Admin office phone is 3501 6428