Welcome to National Poetry Month!
I'm not one for poetry - I usually don't "get it" or can't get my syllables to end up the right way without a lot of hard work. Still, it's good to know a thing or two about the different poem forms.
I think it was the February 2008 issue of Writer's Digest that introduced me to senryu which is like haiku but the focus is on human relationships and psychology instead of nature. I haven't written one yet (that's a goal for me this month) but I have some links for you to view examples and get a better feel for how they work. The poems are three lines and don't seem to have any criteria for how many syllables need to be in each line.
Simply Haiku has an entire section on senryu.
Senryu Entry Page gives a brief explanation and offers examples.