DIAGNOSIS Those are carbon fines—tiny particles of activated carbon from inside the filter. It's not dangerous, but it's annoying. It happens when a filter isn't rinsed well enough initially or if it gets jostled or damaged.
FIX Take the filter out. 1) Rinse it under cold tap water for at least two full minutes, making sure the water is flowing *into* the open end and out through the mesh sides. 2) Gently shake and swirl it while rinsing. 3) Reinstall, fill the reservoir, and discard the first two full pitchers of water. This should flush any loose particles out of the system.
IF THAT DIDN'T WORK Inspect the white mesh sleeve on the filter for any tears or holes. If the mesh is damaged, carbon bits will continuously escape, and you'll need to replace the filter.
HEADS UP Always handle the filter by its ends, not by squeezing the mesh middle. Squeezing can force carbon particles through the mesh and into your water stream.