I've written before out the poetry we memorize, but every week we read poetry as well. Last year (for second grade), I read selected poems from one book (whatever struck my fancy that day). I would often use the same book a few weeks in a row or pull it out again a few weeks later. The girls would almost always want to listen as well.
I explain any words I think they might not know and answer any questions they have, but I do not use this time to teach about meter, rhyme, or rhythm. I suppose that day will come eventually, but for now we focus on enjoying poetry.
I personally feel wonderful illustrations are a great asset when reading poetry to young children. They will often ask for particular poems just because the illustration intrigues them. All of our poetry time books are illustrated and I encourage the children to pore over the pages during and after our readings.
Eric Carle's Animals Animals compiled by Laura Whipple, illustrated by Eric Carle (of course) - This is one of our favorite books of poetry. We seem to read from it every year and even started third grade poetry by revisiting this old friend.
Poetry for Young People: Emily Dickinson edited by France Schoonmaker Bolin, illustrated by Chi Chung - The children are a little young for Emily Dickinson, but she is one of my favorite poets so I always try to read a little of her aloud each year. Even if they don't understand the poems yet, I hope they will grow to love and know them over time.
It's Raining Pigs and Noodles by Jack Prelutsky, illustrated by James Stevenson - The kids love this book of poetry my mom picked up at a thrift store. They laugh and laugh whenever we read from it.
When We Were Very Young by A. A. Milne, illustrated by Frank H. Shepherd - This is simply a delight for children. Please share it with all the little ones you know!
Now We Are Six by A. A. Milne, illustrated by Frank H. Shepherd - Again, simply perfect. These two volumes by Milne are the only ones I think we read from cover to cover.
Bright Star Shining: Poems for Christmas edited by Michael Harrison and Christopher Stuart-Clark - I selected some from this to read during Advent.
Paul Revere's Ride by Henry Wadsworth Longfellow, illustrated by Ted Rand - This is my favorite illustrated version of this Longfellow poem. It may not be historically accurate, but it's fabulous poetry and children love to hear it read aloud.
The Bill Martin Jr Big Book of Poetry, poems selected by Bill Martin Jr., illustrations by a variety of wonderful children's book artists - I liked this book so much it's the one book of poetry I purchased for us to read aloud in third grade. First Son can often identify the illustrator for the poems (by naming other books by the same person, not necessarily by name).
A Child's Garden of Verses by Robert Louis Stevenson. Our copy is illustrated by Eloise Wilken and other early illustrators, but this Tasha Tudor one is probably wonderful as well. No child's poetry collection is complete with Stevenson.
Read-Aloud Rhymes for the Very Young compiled by Jack Prelutsky, illustrated by Marc Brown - I've had this book since First Son was a baby and I still enjoy reading from it to the children.
Tomie dePaola's Book of Poems compiled and illustrated by Tomie dePaola - This is another book of poetry we really enjoyed.
Poetry for Young People: Robert Frost edited by Gary D. Schmidt, illustrated by Henri Sorensen - The children are a little young for Frost as well, but as he's one of my favorite poets, they'll have to listen to it a bit! I went to college in New Hampshire and Frost's poetry always reminds me of those busy and carefree years. (At least, looking back I think they were carefree!)
Poems and Prayers for the Very Young selected and illustrated by Martha Alexander - I've mentioned this before as one of my favorite selections of poetry for young children. I'm so sorry it's out of print. The illustrations are so sweet and many of the poems are perfect for little ones to memorize and sing. I'm biased because I remember this book from my own youth (though not the copy we own).
I love the smiles and excitement I see in the children when they recognize a poem, especially if it is one they have memorized and can recite along as I read. It's as if they are greeting an old friend.
This year, in third grade, we are continuing our poetry readings once a week, but I am also trying to incorporate poetry into our American history. I have a couple of anthologies and we'll be reading a wonderful biography of Walt Whitman alongside our study of the Civil War and Abraham Lincoln's assassination.