Actually that's not quite right. (I was just talking to an engineer who services nuclear reactors for a living about this.) Most of the spent fuel rods are left sitting in storage pools ("wet storage") next to the reactor. What you refer to, cask ("dry") storage, is the other option, but is not as prevalent due to cost and opposition to storing nuclear waste outside. Ironically wet storage is the less safe of the two, because mechanical failure could result in the coolant boiling off, at which point radiation would be released into the air. Apparently this scenario is outlined in one of those Discovery channel, "What would the world be like without humans" shows.
As I recall, wet and dry storage are not mutually exclusive. The usual scheme is to let the waste sit in wet storage until the shorter-lived isotopes have decayed away and the waste has become cool enough to move to dry cask storage.
