The most difficult task facing singers who aspire to sing polyphony is learning to sing without the support of instruments. A whole range of skills that have been in the background must emerge to the foreground, such as the ability to sing an independent part, stay on pitch without outside help, count and render rhythms in a variety of tempos, blend sound and words with others, and other factors. Half the battle is working through the psychological realization that you must produce the music itself and not merely attach your voice to the percussion instrument nearby. Singing polyphony can be an completely new challenge even for experienced singers of modern music.
Fortunately, Orlando Di Lasso came to our assistance in the 16th century with a series of duets designed to teach all these skills. They are wonderful pieces of music in their own right, even useful for liturgy.
Many of these "Bicinia" are available here. You can download them and practice them with one other person. You will amazed at how challenging such seemingly simple music can be.
Another set is available here but because the software used to create the files is no longer supported, these 7 duets are available in a single download from Ceciliaschola.org.
Prepare to spend the next six months perfecting them, and, at the end, finding that you are a much better singer and reader. Thank you Di Lasso!
By the way, if anyone knows a source that discusses these treasures and the circumstances that lead to their creation, please email me.