Common theory is they do, in fact, drop on the same timer. From what little testing I've done that seems to be the case. (Provided after 2 hours of seeing mostly rites I left the area for a lower level one.) Another theory states if a rite can drop it will drop first, In other words once one drops you have 5 minutes to find a geode or the rite will overwrite it. This can be bothersome on the bigger monsters outside abyssea (It's usually a bad idea to farm in there anyway) It can take much longer to kill high level monsters for that drop. Bottomline is in either case go to low level areas unless you are intentionally farming both.
The rate geodes drop haven't been an issue for me, its the stupid timer which eats up all my time. Provided I'm sure in the name of the great balance god, should the timer be removed the drop rate even in weather will have to be cut to 1/5. One final note, rites have 0 crafting value so for the most part they are nice and cheap (breeze excluded), I think the reason geodes haven't followed suit is Pyrosoul type rings have gone up in price, and these new trials take roughly 3x the previous amount meaning the trial weapon creators have likely not finished yet.