Discourse intonation attempts to explain how intonation patterns in English affect the communicative value of speech, through the use of falling and rising tones along with changes in pitch. The teaching of intonation seems to sit naturally with communicative language learning, but it is not an easy aspect of English to incorporate into the EFL classroom. Discourse intonation, with its emphasis on communicative value, is appealing theoretically for instructors and for practical reasons (aiding in understanding naturally spoken English) for learners. This paper reports on a study conducted at a language school in Japan, which aimed to establish a balance between the aspects of discourse intonation that could survive in the classroom and those that would help students to better understand spoken English. The study finds support for the teaching of some features of discourse intonation but also suggests that some features are too subtle to survive in practical teaching.