I guess it depends on your definition of clean. I use the classic Bodum French press, so your mileage may vary (some cheap presses catch more grounds in the screen area).
I wash the glass carafe like any glassware, and then simply rinse and wipe the press itself under the tap without soap throughout the week. Once every couple of weeks, I’ll dismantle the plunger and thoroughly clean it with dish soap to remove any stains.