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It looks very squarish/flat so making a cutout for it to sit in shouldn't be too much work for a router. Then small chisels to finish.

The wood mine was glued to looked like Cedar. Like I said, Like an old fence board. It was an old piece of wood and you could tell it had been around water a lot of its life.

I don't know what is traditional, but i got a Jnat once that was glued to a piece of board that looked like a fence board. Not fancy. Now i just make my own boxes out of whatever wood i feel like.