I've read about dehydrating cut flowers in cat litter. Of course, you have to completely envelope the flowers in cat litter, so perhaps it's just a matter of getting enough litter into your apartment.
Perhaps you could contact a construction company and have them bring you a whole dump truck load of cat litter which you could then shovel into the appropriate rooms.
If you do try this, we'd love to see pictures--and of course, do write up an ongoing report about it, too. You know, give us the play-by-play of how it's working and all!
P.S. Better leave the kitchen unlittered, though, otherwise you won't have enough moisture around to wash dishes! (But, hey, drying them will be a snap!)