Japanese poetry is very unique. It does not depend on rhyming as much as Western poetry forms. I enjoy this because I am not very good at rhyming to begin with.
The two styles that I like experimenting with are the haiku and the tanka. The haiku is composed of three lines with a syllable count of 5-7-5 for a total of 17 syllables. It must contain a seasonal word. Otherwise, it is called a senryu. I have used it as a tool to write science fiction haiku, aka scifaiku.
Here is an example of one that I wrote several years ago:
Black haired girls
Riding on red black hoverbikes
Travel New Tokaido
Tanka is similar except that it has a longer syllable count: 5-7-5-7-7.
Here is an example that I composed just for this blog (don’t you just feel so special inside?):
I feel bitter now
Circling a black world
That was once my home
I am but a ghost orbiting
Waiting to join my family.
It is a great form to learn if you want to try something outside of the conventional forms taught in many schools. Give it a shot.