Why do these apps "like to access the microphone"?
It's a new permission that Apple added with iOS 7 to let you know which apps need to access the microphone.

Now, for those apps that do not relate to the microphone, my guess would be one of 2 things:


Developers of such apps did not update their apps to iOS 7, so the OS needs to ask for your permission on accessing the microphone, or
Developers did update their apps to iOS 7, but they did not address this API in their code yet.


Either way, we should see less of these pop up messages as iOS 7 becomes more mainstream and developers get the time to update their apps fully.

In my experience, not a single app "broke" for denying it access to the microphone, so it should be fine to hit "Don't Allow".