Unfortunately Flash is installed in different ways on different browsers. I have the whole CS suite, so it might be possible there is no Flash player installer in the basic DW installation. What you need to do is as DW is telling you and download the Flash Player to install. IE uses ActiveX to install Flash and Chrome has it's own and Google updates the Flash player with its browser which may not directly correspond to the Adobe updates.
If you go to http://get.adobe.com/flashplayer/ in IE click on the "Different Operating System or Browser?" link at the top. Click on your operating system which I will assume is Windows because the Mac only has one installer and you don't strike me as a Linux or Solaris user. Then select "Flash Player 10.3 For Windows - Other Browsers". Run the installer and you will be good to go.