I occasionally write sonnets (for the birthdays of dear but absent friends), both in the Shakespearian and the Italian schemes. But other than that I rarely if ever write poetry that deliberately rhymes. (A couple of lines in a poem may rhyme by coincidence, but that doesn't really count.) I find that writing in blank verse provides useful structure, but rhyming is hard enough for me to not be worth it except when I'm deliberately using a form that calls for it. (I make heavy use of a computer program I have that serves as a rhyming dictionary.)
I suspect that I might have started out in heroic couplets (rhyming pairs of iambic pentameter lines) rather than blank verse if I hadn't read enough Alexander Pope my senior year of high school to become heartily sick of that form.
Speaking of sonnets, there's a third kind of sonnet that's rarely done, the Spenserian. Its rhyme scheme is ababbcbccdcdee---like the Shakespearian form, only with the first rhyme of each quatrain being the same as the second of the previous quatrain.
And one correction (in the "it's not quite that simple" department):
Kiev Shawn wrote:
One of the more complex rhyming schemes that comes to mind is in a form called an Italian sonnet. It goes abbaabba cdcdcd.
That's 
one of the possible rhyme schemes for the Italian or Petrarchian (Petrarchan?) sonnet. It's divided into an "octave"---eight lines---and a "sestet"---six lines. While the octave is always abbaabba, there are a variety of accepted variations in the sestet's rhyme scheme. I've most often used cdecde, but there's also cdcdee and ccddee, and probably others.