No. There are only five, six, seven or so vowels in Japanese and most syllables are open. There's little point in rhyming. Imagine a person every sentence with あ. That sounds a bit funny, actually. Therefore, most popular music in Japanese (be it enka, kayokyoku or J-Pop) don't rhyme.
I should mention that many languages do have very few vowels, but Japanese has much more open syllables, i.e. syllables end in vowels. Also, Japanese syllables are rather scarce. A combination of these factors - not just having few vowels - contribute to the lack of rhyming.
The 5-7 rhyme, reminiscent of haiku and tanka, is prevalent, thought. Perfume's song "575" does its lyrics in haiku. I have not paid much attention to alliteration, but I remember of seeing some.