Usually intel motherboards are among the best in terms of power consumption, my download station in my signature can idle close to 10W, but im running 3 fans, probably can reach lower if you use less. Most of the builds that you can search here reach low power consumption have used intel motherboards, so thats my suggestion.
Idk really what you going to use it, so idk if an atom would work out for you or not, a motherboard that has been used in low power consumption projects that has intel 2 nics already built in is the Intel BOXDQ77KB LGA 1155 Intel Q77 HDMI SATA 6Gb/s USB 3.0 Mini ITX Intel Motherboard
, it also has a PCIe 1x so you can get your desired 3 lans connections, the problem is that i think is no longer in production, and there is limited availability on the market thus the price is higher than it should, but i bet you can find some in ebay. Check Fluffy2 - 5.9 Watt high-end desktop computer
, he uses the same motherboard and manages 5.9W with some modding, still i think with a good brick and few fans, you probably can lower below 10W easily with it, now adding the nic will increase the consumption though.
As an alternative, check SPCR Gigabyte GA-H77TN Thin Mini-ITX Motherboard
, they manage similar consumptions to the DW77KB, there is even charts there where you can compare them, on the downside no dual nics but you could go with the twin lan port cards.
Good luck with your choice of motherboard, and please share your results, its always interesting reading on very low powered pcs.