- In some rare cases, I've seen Steam versions of games not work when instinct told me they really ought to be working (if they had pretty low-spec graphics requirements, for example). If a game has a demo version available as a stand-alone .exe, try that; if it works when the Steam version doesn't, email the developer! In my experience, they're often receptive to feedback and willing to help more people play their games.
- If a game flat-out doesn't start, and you're feeling determined, you can try to run it from the terminal to see if it prints any intelligible errors. To do this, close Steam, open a terminal, and (on a default install) run:
wine ".wine/drive_c/Program Files (x86)/Steam/steam.exe"
Steam will reopen, printing miscellaneous output to the terminal. Try to run the game in question, and if you're in luck, when it crashes, it may mention missing a specific library that you can install through winetricks or track down elsewhere. If there are no clues, my advice is to forget about it for now--you can bash your head into a wall all day, and if it's thanks to a bug in Wine or your video driver, you likely won't be able to do anything but wait for a fix.
- The Wine website is a great resource for tracking compatibility with various games in various versions of Wine. As always, they'll be able to support you best if you're using an Nvidia card with an updated version of Wine (in which case you're less likely to have issues in the first place, but that's sadly how the cookie crumbles). If you're feeling confident and generous, please publish your own compatibility reports!
Hope that helps. If the wind is in your favor, you'll be gaming on Linux in no time, and you might even find that you don't miss Windows at all. Welcome to the world of free software! You are now a socialist with a very short temper.