You can listen to all these podcasts off their websites or find them in the podcast app on your phone. iPhone has Apple Podcasts by default; Stitcher is a free podcast app for all phone types.
"Every weekday, U.S. Poet Laureate Tracy K. Smith delivers a different way to see the world – through poetry." A great selection of poems and a model of how to talk about poems, both their structure and meaning.
Example:
"Black cricket in the doorway, on the ceiling, in"
"Shares the stories behind the poems and features interviews with contemporary poets." Not currently recording, but past episodes are hilarious and interesting conversations about poetry. Decidedly non-stuffy.
Example:
"Featuring Nicole Shante"
Several different podcasts, from readings of poems to discussions of poems by poets, critics, editors, and other literary people.
Example:
"Things to Do in Hell"
"Dead To Me: A discussion of Claudia Rankine's 'Don't Let Me Be Lonely"
Six of the Best Poetry Podcasts Review these suggested Podcasts about Poetry.