A Roofing We Will Go. . .
The best 2 types of "natural" roofing you should consider are Cedar shakes or Birch bark shingles.
Splitting Cedar for shakes is relatively easy (but it takes a lot of Cedar trees to get enough shakes. . .especially for a barn roof), or cut down and peel the bark from Birch trees. Weigh it down so it lays flat and let dry (about a year) then it could be cut into shingles and used for a roof.
An earth roof needs to have a super sub-structure to hold up, and you would need Cedar or Redwood or a similar type of "rot resistant" wood to hold up to the damp soil or moss or what ever type of "earth" type material you use. And your rafters would have to be placed no farther than 12" (o.c.) apart to support the weight. It would also be advisable to use chicken wire under the earth roofing to help support and distribute the weight of the (moss or grass) roofing.
JM2C! :)