Wood has it's own undeniable charm.
The wood would need to be treated against lice, mites, insects and the elements yearly.
If the wooden structure is low in quality and is not sufficiently protected, you may find that it will need to be replaced after a few seasons. This can become a costly adventure.
Plastic or metal
These are easily cleaned just by using a good power hose and will last for many years. Metal coops do however need to be galvanized and protected from the elements to avoid rust. Both need to be in shade throughout the year to avoid overheating or the cold.
Stone or Brick
Crevasses on the inside need filling with a bird friendly material to avoid insects finding refuge.