Same thing here... I've had this problem some years ago - it has something to do with some old movie codecs I think... the solution is simple and easy: Go to "Data" folder and delete or change name of the movie "Sierra.avi". You'll get to the main menu this way. I'm not sure if there are problems with the other videos as well however.
I can't understand why GOG didn't solve this problem in their installer...
You can have a look here, the problem is described in detail: http://archive.adventuregamers.com/forums/showthread.php?t=19291
As you can see, it's a very old problem.
Sorry to resurrect an old thread, but I just had this problem, and this fixed it. So if others are also having this problem, they will find this thread. Thank you so much
Be warned: I'm not sure if this solves the problem completely. There are some scenes later in the game, where GK3 uses other AVIs ingame. What really solved the problem for me was uninstalling some additional codecs I had installed - I think it was FFDShow or some other big codec-pack. After that everything worked fine, including the sierra-logo at the beginning.
Have fun, it's a damn fine game. :)