You can have the world keep track of which actor's "turn" it is, and have the world's act method call a method on that actor. You'd also have a method in the world so that actors could signal that they have finished their turn. The act method of actors would not be used (or used only for simple repeating animation).

thanks for replying. I think I understand what you're trying to say but I'm a beginner in Greenfoot and I'm lost half the time. Could you give me examples in code please, if possible?

For starters, create an actor class called eg "TurnBasedActor" and have your other actors extend it. Have a method called 'doTurn()' in it (which does nothing). Your other actors can re-implement this method to do whatever they want. Then, have a variable called 'currentActor' of type 'TurnBasedActor' in your world. In the act method of your world, call currentActor.doTurn(). If you need more help, please ask specific questions rather than just a general 'write all the code for me'.