There's a great beginning reader book, available to go with the song. It doesn't match perfectly, but is nice to use in combination with teaching the song. The more we can do to support literacy in correlation to music, the better!
I first sing the song to the students, having them listen to all the things I'm going to do. After we list all the silly things, we brainstorm ways to show through movement the actions described (dabbing heads with glue, tying up in string, etc.) I sing it again, with them moving to the actions described in the song. Then, I teach them the chorus by rote, if they haven't already learned it from hearing the song sung a few times.
In future classes, I teach them the verses, and we add student lead movement to all parts for the final performance.
I was first introduced to this song when Peter and Mary Alice Amidon visited my school a few years ago for the first time as Artists in Residence. (I highly recommend them, by the way!) Mary Alice sang the song to the students. It is on their recordings as well, and is a more gentle version that Woody Guthrie's.
Enjoy Pete Seeger performing Woody Guthrie's song for his children!