Stefan, what dictionary did you look in? Here in the US, I've always heard it pronounced like Gee-Nee. The i's are not pronounced separately.
I looked it up in two online dictionarys: www.m-w.com
Both agreed with me.
I also looked it up in the online Oxford English Dictionary. That didn't help me with the pronunciation, but it made me realize that I didn't completely understand the word. Genii is the plural of Genie or Genius (but not the "smart person" meaning, but the "spirit" meaning of Genius, which I didn't even know existed until just now.)
Definition two of "genius" from the OED: 2. A demon or spiritual being in general. Now chiefly in pl. genii (the sing. being usually replaced by GENIE), as a rendering of Arab. jinn, the collective name of a class of spirits (some good, some evil) supposed to interfere powerfully in human affairs.
I've never heard this definition of Genius before. Fascinating stuff.