By the way, in my country, there's a saying that in the spring, when you first hear this bird singing, and you have some coins in your pocket, this year is going to be financially successful for you. This saying obviously originated BEFORE credit cards and PayPal, though. How cuckoo deals with the advancement of technology, I do not know yet. Another saying about the cuckoo is that you should count the number she sings her minor third interval. This will indicate the number of years you are going to live...
Another sound frequently imitated is the hunting signal symbolized by the hunting horn. Since the horn didn't have valves until the 19th century, it could only play the natural overtone or harmonic series affected entirely by the lips (from C: C-C-G-C-E-G etc.). In orchestras, very often these natural hunting sounds would be played by the pair of two French horns.
Today's piece for sight-reading is Voluntary I from 10 Voluntaries for Organ or Harpsichord (1758) by John Bennett (1730-1784), English Baroque organist and composer.
Here is the score for playing. You can hear the sounds of the cuckoo in m. 164-182 and the hunting horn in Trumpet and echo passages throughout the 2/2 section. Although the meter is either 3/2 or 2/2, I recommend you play slowly and count in quarter notes for accuracy.