When I started my grad program at Eastman, I taught third-year aural skills (what we call Musicianship at DePauw, called Ear Training at other schools). Back then, the entire third year of theory and aural skills was devoted to 20th century music. In aural skills, the challenge was to find a means of helping the students to sing atonal melodies. The system that Elizabeth Marvin – director of the aural skills program at the time – developed was an intervallic numbers solfege. Students sang the intervals of the melodies by the number of half-steps within the given interval. Thus a major third was "Four" and a major seventh was "lev" (short for eleven). The problem was that these students had been trained for the first two years to associate numbers with specific pitches in a tonal context. Now a given pitch could end up being sung with two different numbers in short succession, because each instance had been preceded by a different interval. We ended up scrapping the system pretty early on, with no real satisfactory substitute to be found.
The pitch charts described by Anne-Carolyn are a possible solution. An important step is to use a variety of different notes as pedals. This could be a fascinating way of analyzing a passage, as it can delve into all the possible ways of aurally comprehending the melody. I do think rhythms should not be left off, though. I would slow everything down so the majority of notes would be beats at about 60 bpm (slower for beginning students). But rhythm is so important in the character of a melody, especially in deciding which notes are resolutions, that I could not see doing away with it altogether, unless the student cannot proceed any other way. I will pass on this suggestion to my colleague who will be teaching Musicianship IV this spring, and see if she finds it useful.