Concept Sets

There are 96 pre-determined sets of cards. Most sets are mono-colored, consisting of 4 cards that fit together thematically or have strong synergy. There are usually some early and some late game cards, to ensure a proper curve.
There are some rare dual-colored sets, which contain 2 dual-colored lands of the correct combination to enable playing these cards as well as 4 playables that fall into one or both of the colors.

Example Set: Green Landfall

This set contains only landfall payoff, which synergises well with other sets containing cards that allow you to play or fetch additional lands.

Packs

Each pack contains 4 sets. There cannot be more than one mono-colored set per color in one pack, although colors may be repeated if the pack contains a dual-colored set.

Drafting

This cube can be used to draft with 4-8 players. Draft 3 packs, passing left-right-left as normal, and picking one set from each pack. This means no player gets more than one set from each pack, but even with 8 players, every combination of two players get some overlap in their packs.
After 3 packs have been drafted, each player has 12 sets of 4-6 cards, of which they choose 6 to form their deck, adding any basic lands as normal.

Goals Advantages

My group drafts approximately once every 2-3 weeks, although not every member can attend every draft. Additionally, most members don't play much or any MTG outside of these drafts. This means the general level of drafting and deckbuilding is fairly casual. This cube concept is meant to facilitate faster drafts that still result in powerful and synergistic decks, even when drafted in a more casual way. At the same time, skillful selection of sets should still be rewarded by cross-set synergies.

Potential Problems

Since packs don't go around the table nearly as much as with regular drafting, there is much less opportunity for reading other players' picks. However, in a casual setting, this probably doesn't occur much anyway.
Re-setting requires sorting all cards back into their sets, which may be annoying, but should be managable.
I haven't come up with a way to build packs in the way described above yet without giving the person building the packs any additional information, but this may be solvable.
There needs to be some way to easily distinguish which cards in each pack belong to which set. Color alone is not enough, as there can be multi-colored sets overlapping the colors of mono-colored sets.
If sets are of different sizes (4 or 6), packs will also be of different sizes. I don't know how noticable this would be while drafting, but it could be an issue, allowing players to see multi-colored packs coming.
Both issues listed above could potentially be solved by not drafting the actual cards, but rather representations of the cards listing what each set contains. However it may be difficult to represent 4-6 cards on such a list without making it unwieldy or lack information.