For the wood slats - if I were you, my choice would be based upon three key factors:
-How much am I willing to spend?
-What kind of weather exposure will my grill get?
-What resources do I have available for the wood?
If the thing will sit outside, exposed to a lot of rain, Pine may look nice for a while, but without regular maintenance / varnishing / painting, it will tend to rot.
If you have access to some, I would recommend Teak. It is strong and rot-resistant - once you get some, you would just need to sand it and periodically apply a natural penetrating oil, which would also look nice.
Marine Plywood of sufficient thickness, cut into strips, with a nice veneer, may also look and work nicely.
If you have any cabinet shops or marine / boat restoration people in your area, if you ask them nicely, they may be able to point you to somewhere where you could get some suitable wood / maybe they could sell you some scraps that you could cut-down, if you have access to a decent table-saw.
Just throwing-out some ideas for you to chew-on.