@tinar I had exactly the same problem and was about to give up.
However, after deleting (not just deactivating) the WP Super Cache plugin it worked like a charm.
PROBLEM SOLVED!!!
The important thing is that you enter an e-mail address from your domain in the From field in the Mail section of the Contact Form 7 Settings page.
Many ISPs will not allow you to send mail unless the From e-mail address is the same as your domain’s.
For example, enter anything@yourdomain.com in the From field, if your domain is yourdomain.com.
This solved it for me. I tried WP Mail SMTP and fiddling with the php.ini and wp-config.php settings but it didn’t work and it wasn’t necessary. Switching to other Contact Forms didn’t help either.
