The man responsible was Japanese composer Akira Ifukube, who wrote Godzilla's iconic music themes and scored most of the Godzilla series. In making the original Gojira in 1954, Ifukube and his team had very little time to write the music and create the movies sound effects. For Godzilla's roar, they first looked at real animal sounds like lions, tigers, or elephants. These proved unsatisfactory, however. Godzilla was an enormous monster beyond our imagination, so it stands to reason he should have a wholly unique and unearthly sound. Ifukube hit on an idea to use a stringed intrument, an inorganic sound. He used a contra bass( double bass), one of the lowest pitched stringed instruments in the world. He actually had to borrow one of the rare instruments from the Japan Art University's music department. He loosened the strings and ran his hand across them with a leather glove. They recorded the sound and played it back at reduced speed to create Godzilla's iconic roar. He also created Godzilla's thunderous footsteps with a primitive amplifier.
Unlike American movie monsters, which used animal sounds, this technique of using metallic, or inorganic sounds would become standard practice for Toho's monsters, including the likes of Rodan, Mothra, and King Ghidorah. Notable exceptions are King Kong and King Caesar, both mammalian monsters, which used elephant sounds.
Of course, Godzilla's roar has been updated for his new appearance in 2014, but the base is the same. Below are a couple of related videos. The first is the full roar from Godzilla 2014. This is the scene in Chinatown and his roar is a challenge to the other monster, Muto. It's so loud and powerful and sooo long it goes on forever. You feel like he's going to swallow you whole. Members of the press at advanced screenings said it shook the whole theater! The second video is a mini-documentary with Godzilla director Gareth Edwards and others about how they wanted to pay homage to Godzilla's characteristic sound.