The green success message and not receiving any error messages is a very clear indication that CF7 is sending the email.
So your email is probably being seen as spam somewhere in the email process. This is usually due to email headers (in particular the From email address used) or email content.
Emails can be considered spam for a whole host of reasons and get rejected at many points in the email process in addition to your spam folder.
See Contact Form 7 Email Issues - this links to a much more comprehensive article which also covers emails being considered spam.