We native English speakers often pronounce this name "REE-oo" or "RIGH-oo," but neither is correct. Instead of separating the R and Y, try to pronounce them together: "ryoo."
It just sounds like the ridiculous name of a fan character, usually coupled with other names. It's supposed to be simple and mysterious. I think it's overused for its meaning.
This name is overused in Japanese video games to such a ridiculous extent that TV Tropes actually has a big list of characters named Ryuu or variations. Something tells me that Street Fighter is to blame. Honestly, I'm getting a little tired of hearing this name constantly. It sounds like something an otaku would call themselves, and I doubt it's really that popular on real Japanese men. People really are just using it for its meaning - why not try something with a little more life to it?