Cultural displays in a zoo exhibit shouldn't be roughly shoe-horned in. It should feel organic, like it belongs there. They should be planned for inclusion from the beginning.
One way to do this is to plan exhibits so that they tell a story - one that includes the entire story of the animal, including its place in culture and history. Think of it as telling a story, with the animals alongside with educational displays to share history or culture. For example, imagine telling the story of the Lewis and Clark Expedition - grizzly bears, bison, California condors, and other animals that Lewis and Clark met along their trek to the Pacific would be exhibited on this route. Signage would include quotes from their journals detailing their encounters with those animals. Displays would also exhibits of Native American cultures - teepees, canoes, etc - that were met along the journey and sharing stories of those cultures related to the animals that Lewis and Clark met.
Another example could be something along the lines of "Expedition: Okapi." Zoo visitors could follow the trail of Sir Harry Johnston as he enters the Ituri Forest, searching for the mysterious animal that he has learned about from native sources. As the trail progresses, visitors may encounter more animals from the Central African rainforests - antelope, primates, birds, reptiles - as well as cultural displays of Mbuti (pygmy) culture to discover of these people lived - and continue to live - in the heart of the Congo. A series of clues along the trail, such as stimulated hoofprints going across the path, a piece of okapi skin on display in a "native village," and maybe even a (fake) pile of poop could build up anticipation for the final display - an encounter with a real, live okapi.
Featuring indigenous cultures in a zoo exhibit can be problematic, but it's possible to tell stories about history or culture without doing so. A chimpanzee exhibit could be built around the theme of Jane Goodall and her pioneering field work among the apes. The exhibit could be built to highlight behaviors that she herself observed in the wild (well, maybe not the war and cannibalism), such as tool use using a recreated termite mound. Signage text could be lifted from her journals and logbooks. The visitor portion of the exhibit could be a recreation of her campsite. There aren't many Galapagos species available in zoos, but something similar could be done with Darwin and giant tortoises.
Displaying animals in such a manner does more than make for a more interesting visitor experience. It helps place animals in their proper context - not in a vacuum, one display case or yard among many, but surrounded by a complicated web of life, that we humans are ourselves a part of.