Flushing River still exists, though it is a far different watercourse than it was originally. When the town of Flushing was settled in 1645 along the marshy streams in what is now Flushing Meadows/Corona Park, the creek ran from Kew Gardens (where the old site of the headwaters of the creek is marked by the street Vleigh Place, after the Dutch for Valley).
The marshy land watered by the creek in Flushing Meadows, which today is a green and landscaped expanse—was famously turned into a giant ash dump in the 19th century, run by the Brooklyn Ash Removal company under “Fishhooks” McCarthy.
In the novel The Great Gatsby, Fitzgerald describes how the area looked in the 1920s:
The valley of ashes is bounded on one side by a small foul river, and, when the drawbridge is up to let barges through, the passengers on waiting trains can stare at the dismal scene for as long as half an hour.”
Other than the ashes, swamps occupied the land, fed by the marshy drainage ditch of Flushing River. But in just three years, between 1936 and 1939, the city covered the vast fields of ashes, dug out a new channel for the river, filled in the swamps, created the new Meadow Lake, and landscaped 1,200 acres to create the utopian World’s Fair of 1939.
Robert Moses, who was a driving force behind the transformation, wrote that his teams “leveled the ash mountains, and rats big enough to wear saddles, with white whiskers a foot long, gazed wistfully at the bulldozers and junkies who disturbed their ancient solitary reign.” By 1939 it was impossible to see any evidence of either the ash landfill, or of the natural topography that had replaced it.
For more information, see http://en.wikipedia.org/wiki/Flushing_River. The article mentions that Flushing Creek received water from Fly Creek, Ireland Creek, and Horse Brook; please email me if you know anything more about any of these other watercourses.