If you are hearing "che," as I am, that is a problem with the audio. The moderators just don't have the ability to fix the audio.
чай is pronounced just as it's spelled (rhymes with pie, cry, rye, and high). This word for tea is pronounced the same way in many other languages, and English borrowed it for the Indian-style spiced-tea-and-milk drink that is now so popular in US coffee shops, hipster or otherwise.
(scroll down for audio file)