I'm confused, do you have 3 devices located away from your output (displays)? How are the output devices (TV's I assume?) going to accept the signal and display the media? DLNA over ethernet is one way but it is not elegant from my experience.
HDMI over ethernet data is not the same as what is handled by your LAN router. Your LAN router will have nothing to do with this distribution. As you have three sources that you want accessible in three different rooms, you're looking for some form of HDMI Matrix distribution system.
If your TVs support streaming media from the NAS on your LAN, you can still have your TVs connected to the LAN, but the HDMI distribution will need to be done independently of this.