It's pretty much listen for a voice part, follow the voice part all the way through the song, and write down the notes they are singing. Then, get the next voice part and do the same thing. If you figure them out right, you will see that they form chords. Not necessarily the ones the piano is playing, but chords nonetheless.
Usually, Sopranos will be singing the highest voice part, Tenors will be singing the lowest voice part, and Altos are in the middle.