Hive brings its own board. This is generally considered to be a feature, and I agree. The feature is rare among traditional abstracts, but common among card games.
I tried a straightforward translation to playing cards in my mind. It is possible to lay six cards with their smaller side leading next to a playing card. However, the rectangular form of the cards does not allow for the same patterns as the hexagons of Hive. Different patterns will form. This need not be bad, however.
My question is now, if anyone has ever come up with a working ruleset for something akin to Hive as a card game.
Yes. As an experiment, put paper squares on top of a formation of pieces where the top edge of a square is parallel to the edge of one of the hexes. Lay out all of these squares so that their edges are parallel. You will observe that it forms the pattern airship51 described.
Also see the (sadly poorly photographed) image linked in this old post for a larger example of a staggered square hex map from an old science fiction space wargame (found by forum-searching "staggered squares"): http://boardgamegeek.com/article/4359894#4359894