The ancient Hebrews avoided rhyming poetry for being inferior. Instead, they used couplets or triplets of synonymous lines. Read the Biblical Psalms for examples.
Poetry should be expressive of self: emotions, impressions, actions, etc. With this in mind, rhyming should be thought of as only a tool for expression, not a box into which your poetry must fit.
You could try alliteration, acrostic, other structures, or just write in stream on consciousness, whatever comes as it comes.
I have had my 5th grade students think of an especially intense situation from their lives, then just write their feelings about it as they come. What emerges from this exercise is some of the most amazing poetry I've ever heard of. And it's as real as poetry can possibly be, because that person has lived what he/she is expressing.