I have been trying to search for the Romaji version of "cube" without too much success. There seem to be several variations on the word, although that is true with almost any version of a Japanese word in any form. I have so far seen sanjou, rippou, and even kyuubu. Am I entirely off track?
三乗【さんじょう】 sanjou and 立方【りっぽう】 rippou mean "cube" as in "raise to the third power". If you want to talk about cubing numbers, you should say something like 「４の三乗は６４」, which means "the cube of 4 is 64". 立方 is used in constructions like 「立方メートル」 "cubic meter" and 「立方数」 "perfect cube [number]".
キューブ kyuubu means "cube" as in "a polyhedron with six square faces", e.g. 「アイスキューブ」 "ice cube". I don't think you would use キューブ by itself to refer to the polyhedron; it only seems to be used in loaned compound constructions like 「ルービックキューブ」. Google Image searching for just
キューブ appears to mostly return pictures of the Nissan Cube, for what that's worth.
Other related words: 立方体【りっぽうたい】 also means "cube" as in "a polyhedron with six square faces". In mathematical parlance, you would use 立方体 rather than キューブ. If you want a very technical way of referring to a cube in the polyhedron sense, you can also say 正六面体【せいろくめんたい】, which appears to be a calque of "regular hexahedron". Of course, people don't really say "regular hexahedron" in English, and it seems like 正六面体 is similarly uncommon in Japanese.
So, to answer your title question - which word is "proper" depends on context.