Cover deck - wiring options
You're in a gray area - some inspectors will consider the area under the roof a damp area, some will consider it a dry area.  I think anyone would agree that the exposed part from the ground up is wet.  

I think wiremold will not give you a good result.  I've never tried pulling UF into wiremold but I bet it's a pain.  Bends that are not right angles are a pain with wiremold.  

I think neatly stapled exposed UF would look fine.  If you stapled as much as possible to the top of the beams you'll hardly see it.  The other option would be to do it in conduit, but again with the bends involved you probably can't use ready-made bends, so unless you can bend conduit I don't think it will come out like you want.  

Now from the top of the posts down it's either UF in a sleeve / stub to protect from physical damage, or conduit - PVC or EMT - but it might be all straight runs, so it's much easier to work with conduit.