Progress is reported in understanding physical and mythological elements of the recently reported discovery that handclaps at the Mayan Temple of Kukulkan at Chichen Itza, Mexico produce birdlike descending chirped echoes. Echoes bear strong cognitive resemblance to the sound of the resplendent quetzal, the Mayan sacred bird associated with the temple. ‘‘Picket fence’’ (periodic reflections) from temple staircases explain the presence, frequency, and trajectory of descending chirps. But the rich harmonics are better explained by modeling staircases as acoustical analogues of inclined optical Bragg diffraction gratings. The remarkable conversion of handclap into chirped echo is explained through the time‐dispersive, pulse‐stretching properties of these gratings. Chirped echoes can thus be termed ‘‘acoustical rainbows’’ because acoustical energy is selectively dispersed over time, much as optical gratings selectively disperse colors over space. New evidence strengthens mythological arguments for intentional Mayan use of acoustical features. Echoes may have been exploited at solstice ceremonies. Preliminary evidence further suggests substantial sound reinforcement from the top of the temple to the plaza below. This could explain how Mayan kings addressed large crowds. Impressive flutter echoes and whispering galleries found at the nearby Great Ballcourt suggest possible widespread exploitation of acoustical architecture at this Mayan ceremonial site.