After I installed the update to WordPress 4.4.1, I noticed that I have the same issue with sending messages. Until the update everything worked fine, so I am sure that the update changed something, and that is causing the problem (even if just indirectly).
However, I tried what other users recommended and that fixed my problem!
Until now I had in the From field only the text “WordPress” and Contact Form 7 worked correctly until now, adding the email address automatically to the text in that field.
But now after the current WordPress update, when nothing worked anymore, I figured out that I must also add a valid email address to that field, so at this moment the content of my From field looks like this: “WordPress <[email protected]>”.