Jimmu is, technically, spelled Jinmu. So it is actually pronounced, in Japanese, jeen-muy.Oddly enough, according to http://www.douseidoumei.net/, 3 people in Japan (I'd assume all of them are males) were given the kanji (神武) for Jinmu/Jimmu. However, it's difficult to determine if they're actually called Jinmu or something else like Kamitake, Kanatake etc.
Just to correct the pronunciation that I've put up in my previous comment, I've realised a couple of weeks ago that the 'n' is pronounced 'm' before 'p,' 'b' and 'm.' So, in actuality, it's pronounced jeem-muu (yes, I changed the uy to uu to sort of better reflect the actual sound which, in IPA, is used as ɯᵝ).