Poetry is created with a rhythm. Poets exude rhythm. Every bit of punctuation, every rhyme, every selected iamb is a part of the concoction. Poets strive to create a rhythm and Poetry Interpretation makes an attempt to vocally express it. The trouble becomes when a piece of poetry has too harsh of a rhythm. A poem constructed of four line stanzas comprised of rhyming couplets might be enjoyable to read, but when read aloud the monotony will soon set in.
As a Poetry reader you can still select poems with a rhyming scheme--if you are willing and wanting to learn to read the patterns in a way where you do not fall into a boring, repetitive rhythm. This seems counter-productive to embracing the poet's rhythm construction, but it is doubtful the writer imagined a stunned audience receiving their words. It is kind of like watching Shakespeare being performed by an okay performer and a master. Shakespeare used verse for a majority of his plays and for all of his Sonnets of course. Yet, listen to an okay performer speak either a Sonnet or a soliloquy of verse and you will hear a drumming rhythm being POUNDED out harshly. Then have your ears tuned into someone who understands the words, owns them, and turns poetry into musical speech (here are three examples read by Alan Rickman (brilliant reading of Sonnet 130), Matthew Macfadyen (Sonnet 29), and David Tennant (a collection of Sonnets), great UK actors to represent Shakespeare). The difference is gargantuan.
To break the monotony requires variation, spot-on interpretation, and close attention to punctuation. If you are interested in soldering forward then this article can be of use. There is an easier solution though. And their names are Blank and Free Verse.
To those who are unaware, Blank Verse is poetry that has a regular rhythm yet no rhyme. Free Verse is poetry that has no regular rhythm, no rhyme, or any other form of pattern. Therefore, it is fairly easy to understand why these forms of poetry are often selected. Having less restraint makes it possible for the speaker to have less of a chance of entering into a dull, repetitive beat. The benefit is that simple. However, there is a draw. If you decide to chose a piece of Blank or Free Verse be warned that there needs to be some structure. Without any rhythm the audience will not be able to pick out the musicality of the piece. Further, less structure increases your odds of speaking strangely and without good form.
Poetry success is linked to understanding it and you. You must learn what type of poetry works well with your interpretation and speaking styles. Fail that and you can either become weighted down by strict rhythms/rhymes or go spiraling uncontrollably in a Blank/Free Verse. Poetry is an event that continually tests its competitors. It takes a good ear and one with patience to dominate. Good luck!