For puzzles that use pins and holes, if the holes are drilled accurately, you should be able to use a dowel -inch smaller in diameter than the hole. Buy the dowel stock in any hardware shop. Always use a simple jig arrangement to position the pieces accurately while drilling - never just a pencil mark. It is best to use a special bit with a spur point made for wood rather than the ordinary bits that are designed for drilling metal and will not make a clean accurate hole in wood. Some woods are much easier than others to drill cleanly.
Walnut is excellent.