[gog installers should always be executed with the /nogui option. I assume you have wine 1.6 or newer]

The game works in Linux via wine. Unfortunately it is not perfect, but still it is fairly good.
This setup worked for me, maybe not everything is necessary or there are better solutions.
As usual any comment is welcome.

Download the game setup files and the OpenAL installer for windows from here:
http://connect.creativelabs.com/openal/Downloads/Forms/AllItems.aspx

- Install the game and OpenAL on a clean wineprefix.
- execute winetricks -q d3dx9_36 xinput
- with winecfg add openal32 as Native

It is possible that the game does not detect your wanted resolutions, if it is the case edit the
"My Documents/Codemasters/GRID/hardwaresettings/hardware_settings_config.xml"
file manually.

In my system the game works fairly well, but it is a little slow on exit. It seems GRID is frozen, but actually it just needs few seconds to leave.

Enjoy and do not forget the Bonus codes:
http://www.gog.com/forum/race_driver_grid/bonus_codes

and the DLC content:
http://www.moddb.com/mods/8-ball-prestige-packs-dlc-pc/downloads/8-ball-prestige-packs-dlc-pc

The recent versions of Wine run GRID pretty well without any need to make the hands dirty - everything just runs out of the box. I think, it would be not a big effort for GoG to offer a "Linux version" (as for Flatout for example).