The Kanda River was actually one of the outer rings of a moat system for Edo Castle in the Edo Period, so it’s probably better to think of it as a moat than a river. Train lines run alongside it between Tokyo station and Shinjuku station. The edges of Akihabara (famous for electronic goods and the geek subculture) can be seen in the fifth photo.
UPDATE: The Kanda River was not part of the moat system. It’s a real river. The source is a ways out Westish, about 40 minutes or so on this train line, by the Ghibli museum.