Approach your story as collage instead of narrative. Offer readers or viewers a series of moments, like snapshots in a photo album, and leave it to them to create the connection between those moments. The result will be a story built on impressions and not on narrative. In fiction, this approach has been used most successfully in the short story by writers such as Amy Hempel. In film, this resembles the technique of montage and is usually used for short sequences. But filmmakers such as Terrence Malick have applied a dreamlike rather than narrative logic to entire movies.
Use the logic of surrealism. Like collage, surrealism avoids the straightforward cause and effect of traditional narrative, but goes even further into unexpected and often disturbing territory. An aesthetic movement founded by Andre Breton, surrealism often juxtaposes seemingly unrelated images or moments to create a dissonant but powerful effect. Surrealism also blurs the line between reality and dream, creating a story that is less about narrative and more about the workings of the mind. Noted surrealist storytellers include Jorge Luis Borges in fiction and Luis Bunuel in film.
Explore absurdist storytelling. Absurdism avoids narrative logic as well, but in a way tends toward the philosophical, as opposed to the more psychological approach of surrealism, although writers like Borges combine both. Absurdist storytelling has perhaps been most successful in theater with writers like Samuel Beckett and Eugene Ionesco. In Beckett's famous "Waiting for Godot," for example, two characters engage in a running dialogue that lacks a clear narrative direction.
Experiment with metafiction. Metafiction is another relatively modern movement concerned as much with the process of creation as with the final product itself. Although there is often a traditional story or narrative involved, the real story is the artistic process and takes place in the mind of the creator. A common technique in metafiction is for the author to appear as a character in his own book. John Fowles, Fernando Pessoa and Donald Barthelme all made great use of metafictional techniques. Borges sometimes wrote stories that were reviews of imaginary books.
Draw on older storytelling traditions. Although alternatives to traditional narrative are usually associated with modern aesthetic movements, some argue that certain folk and oral storytelling traditions also operate outside of straightforward narrative cause and effect. The original tales of Robin Hood, for example, were a series of "progressions" without a clear story arc. The modernist writer Paul Bowles was influenced by the Moroccan folk tales he translated.
Organize your story along musical lines. Music offers an alternative compositional logic that is coherent but not strictly tied to narrative. Milan Kundera was the son of a well-known pianist and studied music himself early in life. Though he eventually gave up music for writing, he applied musical structures to works like "The Unbearable Lightness of Being," and within those structures utilized many of the other techniques discussed above. As a young man, Paul Bowles was a noted pianist who experimented with atonal composition. As a writer he often viewed his fiction through a musical lens.