Two issues here... duct work for a range hood normally follows the same restrictions as a dryer. The air is expected to be hot, so you need a sealed metal tube for it to flow through. You can't use plastic or wood or any other combustible or meltable materials. In practice, the air probably isn't much hotter than the ambient temperature of the house, but you have to build it to assume the air could be very hot.
The second issue is that the blower on a hood is not meant to deal with much pressure in the duct. That means the duct needs to be fairly large. A small economy vent hood might get away with a 4" round duct, but they normally use 5" or much larger for a "professional" model. Even large crown molding would only hold a 2" pipe, and even that is a maybe.