To answer your question simply, no. Most games are not designed to work on Linux.
That's not to say that you can't play games on Linux, just that most of the major titles don't
officially support it. There are exceptions, like Heroes of Newearth, the Unreal Tournament series (except for UT3), Doom 3, and several others, which have released and supported official Linux versions.
In addition, there are a number of indie or smaller developers/publishers who support Linux, such as 2D Boy, Wolfire Games, and others who participate in the Humble Bundles. There are also some games, such as Minecraft and many browser-games, which are inherently cross-platform.
But even without those commercial games, there are quite a few original Linux games such as Battle for Wesnoth, AssaultCube, and Tux Racer. There are also many clones like SuperTuxKart, OpenArena, Legends, and Pingus.
And then there's the emulators. There are emulators available for a wide variety of platforms from arcade cabinets to game consoles.
But the real question is, can you play your favorite Windows games. The answer is maybe.
Using WINE (which is a compatibility layer, not an emulator) you can run many Windows applications natively on Linux. When it works, it can really work well. Back in 2002-2004 I used to bring Linux systems to the local LAN Parties and once the games were running other gamers couldn't tell I was using Linux. That said, not every game works that flawlessly. You'll need to check WINE's compatibility list to see how well the games you want to play are supported.
When you're using Linux, you'll also need to be a bit pickier about your graphics card. While the driver situation has improved a LOT since 2004, there are still occasional sticking points with some cards or with specific versions of a certain driver. It's just something you'll want to research before buying to make sure you're going to get the best performance and stability for your money.
http://en.wikipedia.org/wiki/Linux_gaming
http://happypenguin.org/ 
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