Thanks for the tip, unfortunately I don't have anything in my "Additional Settings" and I'm still having the problem.
I found this in the FAQ for the plugin:
The spinning arrow icon shows up once I submit a form, but then nothing happens.
This is most probably due to a conflict between plugins. Try deactivating all other plugins.
Contact Form 7 uses jQuery, and I have installed one other plugin that uses jQuery as well, so there might be a jQuery conflict. The client will be reviewing the site some time today, so I can't deactivate the other plugin right now, but I will investigate further tomorrow.