The pronunciation for 龍 (dragon) is closest to your ree-you as you have written above. But if you pronounce it as two syllables it will inevitably come out wrong, because in Japanese it is a single-syllable word. The "ree" gets contracted with the "you", and the result is a sound that isn't found in any English word. Because westerners find it hard to approximate this sound, it gets corrupted into a more easily graspable ree-you (or any number of mispronunciations).
Note that 龍 + something else might change the pronunciation. The name of 19th-century samurai turned politician Sakamoto Ryōma (坂本龍馬) is one such example. In this case it is something like ree-yo, again with the ree combining closely with the yo to form a one-syllable word, as opposed to being pronounced as two distinct sounds.)