I asked for recommendations to podcasts with poetry and got them immediately. The most simple and straightforward of those is Classic Poetry Aloud (feed). In this podcast each episode consists of a classic poem from English literature read to the listener.
The podcast description of each episode contains the name of the poet, the year of his birth and death (or hers for that matter), the name of the poem and the text of the poem. The episodes will also, usually carry the portrait of the poet. And so, one can look, listen and read and contemplate. The choice is tasteful and the reading done quite well, although a tad too fast for this non-native speaker. I also felt that the reader had a better connection with some poems more than with others, but it surely is far from easy to read poetry.
This formula in its simplicity seems complete and evident (though I'd appreciate if the date of the poem could be added). Nevertheless I entertained thoughts of additional material. Why not add an introduction to the poem, why not let some discussion of the poem follow? But that would alter the nature of the show from pure poetry experience to a cerebral literature show. So what about music, just to get you in the mood? Same problem actually; it would influence the listener's reception of the work. And so, as pure as it is, it should stay. And the listener should be aware, should prepare for listening. And perhaps loop the episode. All in all this is quite the podcast experience.