It's hard to describe who's tune this is and I can only refer to it as the 'commonly used' or 'popular' tune for birkat hamazon. It seems like it is compiled of many unrelated tunes. At times there are melodies for a specific phrase and at other times there are melodies for paragraphs.
Was there an 'original tune' that has now developed and where was it from?
[I'm interested in this, because I dislike the melodies (!) and of course wish to implement universal changes at some very near future.]
Most of the tunes still used were written by Moshe Nathanson in 1938 (some say 1939) for Camp Ramah, a Conservative Jewish overnight camp. He published the sheet music in "Shirei Manginoth", and here is a recording of him performing it.
The irony is that Nathanson was the cantor at the Society for the Advancement of Judaism at the time, the flagship synagogue of Reconstructionist Judaism, where Mordecai Kaplan was rabbi. This is a movement which at the time taught that "God is the sum of all natural processes that allow man to become self-fulfilled." Atheism, according to halakhah, and by the standards of the other movements.
So we are thanking G-d using the tunes of someone who broke away from classical Judaism over a disbelief in what we call G-d.
The common melody for הזן את העולם was indeed written by Moshe Nathanson. There exists a pamphlet containing all his melodies for Bentching, many of which are not in use.
It is worth noting, however, that the general nusach, which comprises most of its modern rendition, is largely traditional to Yekkes and other Central-European communities (Oberlanders, etc.). According to Cantor Isaak Lachmann in Hürben the Bentching was chanted according to a melody quite similar to the current. A very similar melody was used in Hungary in 1898 according to Cantor Mayer Wodak and in Berlin in 1902 according to Cantor Aron Friedmann.
This nusach differs significantly from the traditional Eastern-European nusach for Bentching which is still common amongst Chasidim.
I have heard (but never confirmed) that it was composed by Rabbi Manis Mandel a"h, the longtime Menahel of Yeshiva of Brooklyn.
UPDATE: It seems that this information is at least partially incorrect, as the first part of the tune was apparently composed by Moshe Nathanson (of "Hava Nagila" fame).
I could confirm it better, but based on what my grandfather שיחי told me, there was a small group of Viennese Bochurim (my Grandpa is from Vienna too) that started a small camp in Brooklyn - this was in the early '50 - and they would do Bentching with the tune they have been taught in Vienna. This was one of the first Orthodox camps in the US, and the tune just caught up becoming 'universal'...
