Not sure if anyone here experiencing the problem are on Bluehost, but just in case - I've just found a solution for my Bluehost hosted sites.
The solution relates to making sure that the contact form FROM email is changed from the user-entered value on the form to a fixed email that exists on your Bluehost account. It doesn't matter whether you use Bluehost for your email. I use Google but I simply created a dummy email entry on Bluehost and it worked fine.
So the email from Contact Form 7 will arrive showing that it was sent from that new fixed address. This means that you have to make sure you include the user-entered address in your message body, otherwise you will not be able to reply back.
Now, before I upgraded to WordPress 3.6 I got away without having to do this, but since I've upgraded it's the only way I could get the contact form to work. Check out the full solution here: http://wordpress.shadowlantern.com/how-to-make-contact-form-7-work-on-bluehost/