I was having the very same problem, but only recently. Turns out that the new Dreamhost spam policy was the problem as Somegirlwitha.com suggested. Since it's technically "spoofing" the user's address to send email, Dreamhost no longer allows it.
I fixed the problem by using an email address on the hosting server (i.e. info@mysite.com) in the From field. I then added "Reply-To: [your-name] <[your-email]>" in the Additional Headers field, so that when I reply to an email, it goes to the person who sent it (not info@mysite.com).
These threads really helped me:
WordPress, Contact Forms, the new Anti Spam Policy and why it sucks
Fix Contact Form 7 not working with Dreamhost