It can get worse than that. My problem space has mutable state that is shared between multiple threads. FP's initial answer is "shared mutable state is evil". And they're right! But if that's the nature of your problem, then you're kind of stuck with it.
But the problem with the "recompute and replace an immutable data structure" is that I now have to notify all the relevant threads that they need to replace their reference to the data structure (avoiding race conditions in the process), and that seems at least as nasty as the problems I have doing it the imperative way.