The Michelson-Morley experiment was an attempt, before the theory of relativity was introduced, to find the "absolute" velocity of the earth. It was based on the empirical observation that the speed of light was constant. Using non-relativistic mechanics, this implied that if one bounced a beam of light off a mirror that was moving with the earth and pointed in the direction in which the earth was moving, the time the light took to go in one direction would be slightly increased, while the time it took to go in the opposite direction would be slight decreased, and these effects would not quite cancel, but lead to a travel-time slightly greater than if the earth were standing still. If one bounced a beam of light off a similar mirror pointed in a direction perpendicular to the motion of the earth, the time would also be more than if the earth were standing still, but by only about half as much as in the case of a beam going along the direction of movement. The experiment used interference effects to look for this slight difference in the time the two beams took with the apparatus oriented in different ways - but didn't find any! This paradox was eventually resolved by Einstein's theory of relativity, which showed how the speed of light could be not only constant, but constant relative to any inertial frame of reference.
There is a curious parallel in folk-tale of the Chumash Indian tribe of California. (In copying Chumash names below, I will use "ch" and "sh" for c and s with a hachek, and not distinguish between "i" and "barred i".)
In the middle of narrative 20 (p.152) in December's Child, a book of Chumash oral narratives (collected by J. P. Harrington, edited by Thomas C. Blackburn; University of California Press, 1975) Tupnekch ("child") and Coyote, his uncle, and another coyote (whose skull they found on the trail, and whom Coyote brought back to life) are looking for Liyikshup, the village at the center of the world. The coyotes can travel "by echo": one will give a shout and disappear, reappearing far away (presumably at the place where the echo to one's shout seems to come from).
... "Well, how do we find Liyikshup, then?" asked Coyote. "The only way to find it is for you to go that way, and me this way, and if we return at the same time, that's Liyikshup."
The Tupnekch began to laugh, listening to the conversation. The old coyote told him to stay there where he was, and the Tupnekch said, "But you are very old, you can't run as fast as my kinsman." "We are equally fast," said the old coyote. He added, "Lets have a race to see who reaches that tree over there first!" He gave a shout and reached the tree ahead of Coyote. "All right, now return here," called the Tupnekch. They came back and arrived at the same instant. "That's very good," said the Tupnekch. They traveled along for a while and then the two coyotes shouted and went in opposite directions, but one arrived back before the other. "It still isn't Liyikshup," said the Tupnekch, trying to show that he was as smart as the coyotes. They continued along this way the rest of the day, with first one coyote and then the other arriving back first. They never arrived at the same time. ...
What is this about? My guess is that it is based on the observation that if you are in a valley and shout, then if you hear the echo from, say, the east sooner than from the west, that means you are to the east of the center of the valley, while if the echos arrive at the same time, you are exactly midway. In this story, this has been mythologized into a way of telling whether you are at "the center of the world", where Liyikshup is. Admittedly, this is not based on the subtle computations underlying the Michelson-Morley experiment, and involves traveling in opposite rather than perpendicular directions; but the similarities - in particular, the use of something with a fixed travel-speed to tell whether one is at the "cosmic" center of things - is striking. (The next day the travelers do find Liyikshup, though the use of echos is no longer mentioned.)
Another idea in this story is that of a child raised without contact with human society, like Parzival in European literature. Tupnekch has been raised alone by his grandmother, Momoy (the hallucinatory jimson weed). On the first evening of the journey, the two coyotes start talking about how wonderful sex ('ostus) is, and when Tupnekch asks what it is, they give him a bogus answer, describing the technique for cooking agave, and say "That's 'ostus". When they reach Liyikshup, a couple of young women try to seduce Tupnekch, but he has no idea what they are doing, and hence no interest. They call him stupid. He says he is not stupid, and lists his accomplishments, ending with the assertion that he knows how to make 'ostus, to prove which he describes the recipe for agave.