You can make the grip handles out of almost any hard wood. Walnut is probably the most authentic looking. Remember that Oak contains tannic acid and will turn black if it gets much exposure to moister. Ash has a similar look to oak and holds up to moisture better - it's often used on boats. Hickory is easily found (an old ax handle). Cherry is good too.
I made my first set of handles from Osage Orange, a very hard wood with a beautiful orange tint.

Shaped and sanded, ready for staining.

Both handles are turned and stained and await final measurement and cutoff.
NOTE: After you have the handles made, be careful when drilling the holes. I start a hole at each end. They seldom meet perfectly, but its easier to fix that problem than drilling into the center of one end and having the drill bit come out off center on the other end, or worse yet, out the side of the handle.
Now on to final assembly and the finished product.