Recitation of the Quran without understanding Arabic

I was brought up a Muslim and taught that recitation of the Quran in Arabic (although I didn't understand Arabic and still don't) bears great "sawab". I haven't been able to find anything regarding this anywhere in Ahadith or the Quran. What does Islam really say about this? Is it true or it is just a bidd'a (or biddat)? Technically, it's a waste of time to read the Arabic without understanding a thing of it, isn't it? So what's the logical thing to do in this case according to Islamic teachings?