This had me pretty befuddled, but I found a solution, even if it meant I had to edit the core plugin files. Once I decided go that route, the solution was easy:
In Contact Form 7’s core plugin files, find the file:
wp-content > plugins > contact-form-7 > includes > mail.php
Go to about line # 82 in the file, where you will find:
return wp_mail( $recipient, $subject, $body, $headers, $attachments );
Immediately above that line, add in a new line:
$body = str_replace('<p>', '', $body);
$body = str_replace('</p>', '', $body);
$body = str_replace('<br />', '', $body);
I blogged my solution here.