Contact Form 7 & WordPress 4.3 combo doesnt work

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 spammed somewhere in the email process.
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.

See also Best Practice to Set Up Mail.

I have the same problem: since WordPress updated (automatically) to version 4.3, not only Contact Form 7 but also another form written manually using php to send emails have both stopped working.
They appear to be sent from the user end, but no emails are sent.

Updating Contact Form 7 has fixed that problem, but not the manually written mail.php form.