I almost always take Ancient Race. It has enormous value early in the game. Its value goes down with time, but if you used it well, then you will have a huge advantage in resource production over time. As we all know, this game is about increasing your resource production, and that production increases as a geometric rate. So if you can get a small lead early on, it blossoms into a huge lead later in the game.
It becomes even more powerful in Simultaneous games (ie games against other people): In a simultaneous game, your ship stops when it enters a unexplored system. With Ancient Race, your ship moves happily along since all systems are explored.
So if you want to goto a planet 2 systems over, lets say it takes 1 turn to get out of your home system, plus 2 more turns to get to cross the next system, plus 1 more turn to colonize the juicy planet 2 systems over. (4 turns) Without ancient race, you spend 1 turn getting out of your home system. You sit there for a turn on the warppoint of the next system and see there is nothing there for you. You spend 2 turns crossing to the 2nd system. You sit on the wap point for a turn. Then you spend one more turn colonizing. That's 6 turns. It's only 2 extra turns, but that is alot at the beginning of the game.
And don't forget the fact that you don't have to explore: you can just churn out colony ships and not bother with exploration vessels. That saves you an initial turn or two at the beginning of the game.
And as others have said, knowing where the other races begin helps you plan your initial exploration to maximize the amount of territory you own at first contact. If you see another homeworld near a system with ruins or some nice Huge breathable planets you can send a bunch of colonizers over before the other empire finds them and takes them.
In the PBW games I have played, I've had better quality planets and alot more of them than anyone else by the time I met them. They always ask me what I did. Usually it is ancient race coupled with propulsion experts. In games when I don't take ancient race I don't expand nearly as quickly.
I played one game in an ancient galaxy so there were not many good systems to colonize. The nearest system to my home-system with a colonizable planet was 3 systems away. Because I had AR, I was able to plan for that and send colony ships straight to it instead of wandering aimlessly through asteroid fields and getting sucked into black holes after hitting their damaging warp-points.
Well, I've rambled enough. Suffice it to say I like Ancient Race