Just designed and built a new website from the ground up (at a rather large expense) in the summer of 2013 and we were fortunate that it featured on several 'best of web' sites.
Since then we have found that it has been directly copied once and someone claimed to have designed it - both times we successfully got the sites/references removed.
However, we have just found out that a UAE-based design agency have copied it directly (so much so that 'their' map was a Google map to our office) - the design, images and the entire of our code was simply copied. We contacted their web host with a DCMA request and they took the site down straight away. Unfortunately it has since re-appeared (as a screenshot only, nonetheless it is our site) so they are clearly ignoring requests and circumventing other actions. I can only assume they have moved hosts or are hosting it internally (Whois still says it is the old host but that can't be the case if they had taken the site down once).
Google won't get involved as they say Copyright issues aren't covered in their policy - they don't moderate that sort of claim.