I used Chair Rail for the frame because its flat on two edges to form a nice 90º angle.
I didn't do this but it would have worked out better if I had, go ahead and glue your dowel rod to your chair rail using wood glue. Clamp and let sit overnight.
Measure the outside of one edge of your hex.
Using a miter saw set with a 30º angle cut your pieces.
Find the middle of the dowel rod and drill into the bottom to place your insert.
Put glue in the hole and screw in the insert.
Using wood glue put your hexagon together. I actually just put two pieces together at a time and held them until they were stable (about ten minutes), but if you have a nail gun then use it. Hammering at this odd angle just doesn't work for me.
Once everything is together and dry, use wood glue on the dowel rod and place in your game board.
Use Mod Podge to seal the edge between the game board and frame.
Sand your corners and give it a final paint job or staining.