Among points which have traditionally not been attempted in the modern pronunciation of ancient Greek, the most important, for the appreciation of poetic texts, is the pitch accent. Conversely, it may be justifiable to sacrifice aspirates to a spirant pronunciation and to fudge on various vowel sounds, inasmuch as many of one's hearers, long inured to English, cannot readily hear such distinctions.
Pitch accent, though, is readily heard. Moreover, to pronounce just the quantities when reading verse, with some kind of ictus, appropriate to English or German, introduces a serious distortion. The result of such a practice (somewhat imperiously prescribed by Maas [tr. Lloyd-Jones], Greek Metre [1962:55-58]) is that the underlying rhythm overwhelms the word accents. A combination of quantity and pitch sounds much better than quantity alone. It is also pretty directly accessible to any audience with some knowledge of ancient Greek. A reading of Latin with careful attention to quantity and the ordinary prose accent is likewise better than a purely ictus-based reading.
Besides sounding better, there are also instances in which the original accentual pattern is crucial. An obvious example is Odyssey 9.366. Here, we need to hear Oûtis - let us translate it, with Fitzgerald, as "Nohbdy" - as different from oútis "nobody". If instead Odysseus does not properly pronounce the circumflex, even the dense Polyphemos might suspect some trick - and we need to remember that in an oral presentation he would not hear the capital letter of Lattimore's or Fagles' "Nobody".
Another, more specifically "poetic" example is the opening word of the poem. This is ándrá (not ándra) inasmuch as a resonant consonant (m, n, r, l) functions in Homer like the second element of a diphthong (i or u) to give a trochaic word an extra accent before an enclitic; see West, Theogony (1966: 438-442) for discussion. Unfortunately, most modern texts of the Odyssey (both Allen and van Thiel, for example) ignore the rule. Ludwich 1889 (online version available through TITUS), though, observes it.
Heard as ándrá, the opening word of the Odyssey is correspondingly marked. To be sure, the Greek acute accent was not a matter of absolutely higher pitch, but instead relative to surrounding syllables. Nevertheless, the presence of an extra acute accent suggests a female voice and so combines male and female resonances - the word means "man", but it is spoken with a higher pitch. As the Odyssey unfolds, the point thus adumbrated will be developed in various ways. For example, at the beginning of Odyssey 1.11, as the Muse's voice is ostensibly first heard, we again have two higher-pitched syllables, énth' álloi. In this instance there is nothing specifically Homeric about the accentual pattern; nevertheless, a suggestion that the Muse somehow takes over at this precise point is confirmed by the fact that in the Iliad too the first words following the proem show exactly the same accentual pattern in tís t' ár (Iliad 1.8).
The pattern is also picked up (with appropriate modification for Latin) by Vergil, with the second accent on ándrá being alluded to in his own word virúmque at Aeneid 1.1. Also, tantaéne in the middle of Aeneid 1.11 arguably comes at the precise point that the Muse responds to an introductory request - but responds with a question, somehow "correcting" the preceding proem, just as Iliad 1.8 introduces a question to which Apollo is the answer, rather than Zeus, to whom Homer had referred at 1.5.