I have found dozens, of examples of other narrative structures from the last few decades.
In some cases the structure defines the movie - as is the case in the iconic Memento - in others it is used as a device to make the story more interesting. This is often useful if a lot of ground is being covered - a way of avoiding a sense of a gigantic boring epic. This approach also allows some twists and turns in the plot to keep us gripped. In other cases, it is necessary in order to lead the audience into the right state of mind to really comprehend what the director is trying to say - a good example of this would be Tree of Life. The director Malick has created a dreamy odyssey in which we follow a boy's journey to manhood, charting his loss of innocence, through to his renewed love of the world. This film, which I loved, is only rated 6.7/10 on IMDB and is often dismissed as 'pretentious' and 'pseudo-intellectual' I suspect partly due to the narrative structure.
Non-linear often allows for unexpected revelations which can add emotional depth. It can be playful and also 'jittery' to convey a sense of chaos. This approach is often used in films which tackle memory and self-reflection, giving a feeling of how fleeting things can be and how haunting. Kaufman is brilliant at this, allowing multiple manifestations of reality to be explored in a way which is really enthralling for audiences. He can make very 'ordinary' subject matter seem surreal and intriguing and, in the process, he takes us on a journey of enlightenment.
Films worth mentioning...
L'appartement (1996) - summed up very nicely here: http://www.filmsdefrance.com/film-review/L_Appartement_1996.html
The Usual Suspects (1995) - uses flashbacks and exposition to create high drama and an unpredictable and thrilling conclusion.
Pulp Fiction (1994) - a superb use of non-linear narrative to unravel a complicated plot and tell the story from a variety of character's perspectives. The twists and turns and cause-and-effect elements make this extremely engaging and exciting for the viewer. It is also interesting to apply Propp's Character Theory to this movie and really consider who is the hero?
Short Cuts (1993) - this similarly interweaves several stories and subtly connects a number of people who are themselves rather disconnected from their communities or their loved ones.
Irréversible (2002) - watching the events in reverse order amplifies the tragedy of this story and also allows the director to manipulate our moral judgement. The film is almost unbearably powerful and could not have worked if told chronologically.
Donnie Darko (2001) - has an alternative timeline which muddles dreams and reality and leaves the story completely open to interpretation, not a comfortable feeling!
21 Grams (2003) - the film reveals various time frames for the three main characters in a non-chronological order. Roger Ebert questioned if this was necessary for this movie: http://www.rogerebert.com/reviews/21-grams-2003 "I felt as if an unnecessary screen of technique had been placed between the story and the audience."
I am going to finish here because there are 42 other movies on my list and this might go on forever...