The Daily Mail has a useful map showing the route of the old Fleet River, buried long ago. It is said that you can still hear it gurgling past through a grating in Farringdon Road near the Guardian's old offices. And Diamond Geezer has tracked the entire course. In periods of heavy rain the river, or rather the storm drain that it has been converted into, gives out into the Thames at Blackfriars Bridge.
2016: You want to see the River Fleet? Londonist have the post for you.
2019: The Islington Tribune reported that the River Fleet flows beneath the basement of an extension at the Sekforde pub and it is now being used to generate heating and also to cool the beer. Would we all had a river in the basement!