First off this is a great subject! I love the idea and its a beautiful board, being a player myself. A few suggestions.
1. Get close. I feel like a photo of just a single piece would be better with several others fading away in the back ground.
2. Decrease depth of field. You don't have enough here for the whole picture to be in focus, yet not enough to make it really interesting. If you decrease your depth of field this will bring out the subject you focus on, either a game piece or the die.
3. One lighting source. If you use one strong lighting source that is diffused/not trained on the board you will get a powerful contrast effect that should not cause too much reflection on the lacquer of the board.
These three things should combine well to make an interesting photo and help you to decide clearly what the subject of the photo is. Of course, there are many different ways to do this and many other things you could do, but this is just how I envision these photos turning out!
Other ideas could be compositional, with one side having many pieces left and one side with just one left and the exact right number on the die. Just an after thought I had.
That is a very unusual board, looks antique, and strange that it does not have triangles.
I am a consultant on the game of backgammon and the editor of an online backgammon magazine so I have photographed many backgammon boards and have used many backgammon-related graphics.
I noticed that in the first photo, the dice are well focused as well as the first few checkers above the dice, but that the top two are not quite focused, unless you purposely wanted that effect. Same in the second photo, checkers on the left are focused but not on the right.
You might also try a few shots with checkers of the other color (the beige ones) close by to have contrast.
I usually always take a few shots from the bird's eye view, like this one:
Not such a hot shot, but I think you get the picture.