If your game has a Linux version it's probaby best to run it with the Linux client.
For games only available on Windows, you may be able to run it under Wine.
I was surprised how easy it was to run games. The main issue though is to get your game running flawlessly. It depends really on each particular game. You can search the game on WineHQ but honestly from what I have seen so far the reports are not reliable as people do limited testing and/or on different hardware. I've had some games run flawlessly, and I've had one game run with glitches, despite the solution suggested by WineHQ.
In regard to Rinzwind's copy/paste of the article suggesting a way to copy a game over, it didn't work for me. It seems the downloading mechanism has changed, the main game folder is not created, and the download is done in a temporary folder. I tried copying the files there too, "verify local game cache" etc to no avail.
In fact there is a much easier way:
- Install Wine via Software Center in Ubuntu.
Install winetricks via Terminal:
sudo apt-get install winetricks
winetricks d3dx9_36 vcrun2008 gecko corefonts
This gets you some basics. In WineHQ sometimes they will suggest to run that command to get a few extra packages.
Download Steam installer (Windows version), just double click the .msi file to install it! (when Wine is installed you can double click Windows executables straight from Nautilus and even game shortcuts on Ubuntu desktop)
Now here's the trick: simply backup your game in Windows, and use the restore function in the Steam client (the one running under Wine!).
In more detail: create a backup via (Windows)Steam, copy the folder on an external drive or USB key, log in to Linux, and copy the files again to the Wine Windows, it is located in ~/.wine/c_drive/ That folder corresponds to your (virtual) C: drive, just copy the Steam game backup there. Then run (Wine)Steam and simply browse to C: to find the backup.
You will actually get a game shortcut on your desktop! From there on double click the icon and it should run.
So TLDR, you can have Steam under Wine to run Windows-only games, and use the native Steam Linux client to run Linux native games.
Just keep in mind there is no 100% guarantee the game will run. As an example I ran "Depths of Peril", "Tomb Raider: Underworld" and "Puzzle Quest 2" flawlessly without any prior Wine configuration. On the other hand, despite the "gold" rating on WineHQ I've had some glitches and crashes with King's Bounty Crossworlds running fullscreen at 2560x1440.