But that doesn't mean republics can't have more direct democracy elements in them. For example I love the idea of citizens being able to create a referendum (in a certain state, I believe) in Germany. You still have a threshold, so people don't start creating one for all sorts of crazy ideas, but this is a great example of how citizens can help improve the laws in their countries. I think this referendum is usually added to the next election.
Such system obviously need to be fined-tuned to avoid having the population "abuse" them, but that doesn't mean they shouldn't be pursued. I think they can be great additions to any democratic republic.