The Jumblies by Edward Lear is a fantastic poem to read aloud, I was fascinated by it as a young child - we had it in a gorgeous old book with lovely illustrations by Hilda Boswell but I recently found a modern copy of the poem with equally lovely illustrations
The Hilda Boswell treasuries are fantastic if you can get hold of them on Ebay or Amazon (hardback copies date from the late 1960s so they are often a bit battered but wonderful poems & illustrations)
A A Milne "When we are Very Young" and then "Now we are Six"- just bought these for my just turned 4 and just turned 6 year old. Mostly because I have fond childhood memories of both . Apart from that lots of picture books are rhyming, Julia Donaldson the obvious choice, but Commotion in the Ocean, trying to think of others but can't for the moment!
I would second the Kaye Umansky books, The Spooks Step Out and The Empty Suit of Armour. DS just 6 can read them himself but enjoys me reading them to him too.
I think you can do two things, read them whatever you like, whether they understand it or not, just the sound of something you enjoy will be engaging, rich vocabularly etc etc. Then if you read easier, more child friendly books you can get them to join in a bit more- DD loves filling in gaps for example, I'll read the line but leave the last word for her to fill in. (she can't read, she remembers).
Nonsense poetry works really well here: Edward Lear, Lewis Carroll and Spike Milligan are all favourites. My 5 year old DD really enjoys the nonsense element. Her class teacher used "The Ning Nang Nong Song" as part of their phonics teaching which I thought was really good. Julia Donaldson, Dr Seuss, Giles Andrae, Hairy Mclary also go down well here, as does Roald Dahl's Revolting Rhymes, although it helps if they know the fairy story versions first!
DD was bringing home some poetry reading books from school and found them more tricky than story books, I think because it's more difficult to get the narrative and the metre from a poem until you can scan read.
FWIW, I learnt a lot of poetry at school from an early age and whilst I loathed some of it at the time (don't ask me to recite "Daffodils" by Wordworth ever, even though I still can, bloody wandering lonely as a cloud) I love the space it takes up in my head now. Helps hugely with dealing with Shakespeare later, too. The bits I loved were the fun, nonsense bits and I reckon that's the key.