As in Frank Longo's book, the either/or vowel nature of Y is handled by never having an answer using this letter. All the consonants in the answers for the clues go into the grid; all vowels are discarded. Arrows are provided alongside the box numbers to help orient your entries into this cube; entries "wrap" around the faces of the cube.
Enumerations of the clues are provided as spoiler-text in the comments for those that need a little more help.
Enjoy, and have a safe 4th of July holiday!