You can play as a random race if you do the right things.  Someone posted information about this some time ago, maybe a year or more.  I don't remember who it was or when it was.  That person should get the credit.  I just use it when I feel like playing a random race.

1. Go to your Documents\My Games\GalCiv3 directory.
2. Find the Prefs.ini file and open it in your favorite text editor.
3. Search for Player0Random=0.
4. Change it to Player0Random=1.
5. Save the Prefs.ini file.
6. Start GC3, then start a new game.

This is where things get different.  The Choose Civilization screen will look odd.  The left side where all your races are listed will be scrolled all the way to the end of the list, but none will be chosen.  The right side will not show any information about the race you are playing since none has been chosen.

You cannot play this way.  You need to do the following for every game you start:

7. Choose a race.  It doesn't matter which one, so choose any you can see on the left side of the screen.  If you don't choose a race, the game will not like it
8. The information about the race you chose will appear on the right side of the screen.
9. Click Next, and set up your galaxy.
10. When you get to Opponents, you will see that you are shown to be playing as the race you selected instead of a random race.  Don't worry about it.  You will be playing a random race.
11. Click on Start, and see what random race you are playing.

After you edit the Prefs.ini file, you will always play as a random race unless you edit the Prefs.ini file again. If you want to stop playing as a random race, edit the Prefs.ini file to change Player0Random=1 to Player0Random=0.

Have fun!

Sorry for late reply but I just wanted to say thanks for the help.

I just tested the new 1.8 version and it seems unchanged regarding the randomness. But I don't lose hope

I would like to play total random games with some of my friends in multiplayer. We currently playing Civ V and Age of wonder III and we set everything at random; our leaders, enemy leaders and map type. It is addictive.

I'm going to bump this as well.

It's been over a year now since you said that, pshaw - any chance of some forward momentum on this at some point?

I really do want this feature, it would be very quick and easy to add, and it's been in the single-player lobby THIS WHOLE TIME - really irksome. Just saying.

Haha after posting I dug this one out ... I would also love a "random" option in the choose race screen.  Most other turn-based strategy games (i.e. civilization series) seem to have it ... it adds a lot of randomness and fun to the game in my opinion!