Yes, soon you will be able to play games on your Zune. XNA Group announced that XNA, a toolset that Microsoft provides to developers, will now include development for Zune. This will be possible on the new XNA Game Studio 3.0 which will be made available later this year. What's more, you write a game using the XNA Game Studio and deploy it to all three platforms that XNA Game Studio 3.0 will support: the XBox 360, PC, and the Zune. So now you can enjoy games on the go on your Zune, but it'll lack the 3D graphics to drive high-end games like you can play on the XBox 360.
XNA and Xbox Live teams now have Community Games. It'll be made by community, for community and it'll be published for everyone to play. Get the Game Studio 2 from http://creators.xna.com and start building games for your XBox which can be shared on XBox Live and downloaded for FREE!
Now coming to gaming on Zune, the games will be supported on entire family of Zune media devices that is to say will work on both generations of Zune and you have support multi-player also which will make it more interesting. So just wait for the XNA Game Studio 3 to arrive, available later this year.
Can also checkout this recent video of a winner at a XNA contest game creator, and talk with few XNA people on Channel 9, (GDC08: James Silva, XNA, XBOX Arcade, Zune Gaming, Renegade Dishwasher Samurai Angst and Revenge) which has all the answers to questions like,
what does XNA stand for?
What does XNA encompass from a platform perspective?
What does the recent XBox Live Arcade announcement mean?
Also get to see a demo of a game written using the XNA framework running on a Zune. Here are some pictures from that video of playing games on Zune.