Gee, this seems like such a nice song, but it's a bit of a copyright jungle, and has some interesting political implications. Here is the narrative about the song from the Bear Family box set titled Goodnight Irene: 1949-1953:

The battle over copyright infringements has always plagued the music industry. The Weavers' hit Tzena, Tzena, Tzena was no exception. Joe Jaffe recalls that he or Ernie Lieberman first heard the song in about 1949 through a friend, who had just learned it as camp counselor. They began singing it at People's Artists hoots, where Pete Seeger picked it up.

Seeger later contacted Jaffe at his Bronx apartment and copied the song's lyrics and music over the phone. The Weavers soon began performing it. Before they could record it, however, publisher Howie Richmond researched the song and found it in a book of Israeli folk songs, arranged by American school teacher Julius Grossman. Richmond then had someone (using the fictional name Spencer Ross) doctor the melody. Gordon Jenkins wrote English lyrics, took out a copyright and issued the sheet music.

When the song hit, Richmond received a rude surprise. Mills Music had published the original version, with words by Mitchell Parish and the music attributed to Issacher Miron (a/k/a Michrovsky), an Israeli who claimed he had composed the song. The third part was credited by Julius Grossman. (Indeed, Miron and Yehiel Haggiz had written the song during World War II, while in the British Army, and it had become popular on the European battlefront and among Holocaust survivors.) Mills then sued Richmond's firm, Cromwell Music, and won a judgement concerning the music. Since the latter still controlled Jenkins's popular lyrics, this created a legal dilemma, essentially preventing anyone from duplicating the Weavers' version.
- Ronald D. Cohen

Source: Seeger, Pete. A New-Peaceful-Lease on Life for a Song, 'Sing Out!,' Vol. 43, #1 (Summer 1998), 82-85.