I have inherited a project that has some huge switch statement blocks, with some containing up to 20 cases. What is a good way to rewrite these?
In general, I think you should refactor only when you need to, such as when you want to add more features, but the current design isn't up for the task. You should then refactor without adding the new functionality, and only then add the new feature.
In other circumstances, don't bother with refactoring. Do it when you need to, otherwise there are probably more important things to do.
If you really really need to, then the Visitor Design Pattern is a common switch-case replacement, though you should note it does have drawbacks. (i.e., check out www.objectmentor.com/resources/articles/acv.pdf)