Right click the file and go to properties, then open with tab. Is it trying to open with wine?
You have to understand, Linux is completely different than Windows. .Exe's are made for Windows specifically. If you go to a Mac it'll have no idea what you're trying to do. Linux however has an emulator style program known as Wine, which I'm sure you're well aware of.
Wine was developed to help people who need certain Windows programs adapt to using Linux more and more. Over time, games fell into the mix and now gamers are often running Linux and gaming with Wine.
So, to put it simply, without Wine, you have no games. Period. Done. You need Wine to run the game. It should open, and graphics may have to be calibrated. I personally do not play WoW and have not seen anybody try to install it in Ubuntu to help you with calibrating graphics.
I must advise you, although these forums here are great for getting questions answered, ubuntuforums.org may have a wider range of individuals who could aide you better in fixing this problem. I personally use both forums, but something like gaming with Ubuntu, ubuntuforums.org may be better suited to help you.