Sounds easy. When you mind map the characters, mind-map some of their most striking personality traits and key beliefs. That way, when you see a quote, you know who is likely to have said it.
Example, for the play Romeo and Juliet: "I am fortune's fool." You don't need to have line-by-line read the play to know who said it. Just make a list of the people who could be said to be fortune's fools. Mercucio, who is stabbed in a fit of misfortune, and always plays the fool is one choice. Another choice would be Romeo, who is one of the "star-crossed lovers." Once you've narrowed down the list, your intuition will tug towards the correct answer.
On a totally unrelated-at-the-surface note, check out the book "Requiem for a Dream." It's written for PhotoReaders. The book does not use quotation marks, and never identifies exactly who says what.