Another idea I found on Pinterest. The original source is MeckMom.
My board game boxes use to last no more than two weeks in the room. They got beat up. Quickly. I like how she put all the pieces in little plastic storage bins and labeled the boards. So nice! And it saves space.
I usually put game pieces in these little clear pencil case/bags. Walmart sells them for like a dollar each. I have about 300 (in several different colors). I love them.
Either way you go, plastic bin or bag, it saves space and is super organized.