Not sure if you are still having this problem but thought it would be useful just to add how I have got this to work having just purchased ATV3.
I am on 10.8.4, I have a Billion 7800N ADSL router to which is attached a Time Capsule. I have an Airport extreme extending the TC and my ATV3 is connected to the AEBS.
Like everyone else ATV3 would connect to my iMac when it was not asleep and Home Sharing worked well, when it went to sleep, however, ATV3 would not connect. I had Ethernet and Wireless swtiched on on my Mac running iTunes.
The way that the iMac should be woken when ATV tries to connect to it is via a Wake on Demand 'Magic Packet' sent to the iMac over the network (this emanates from the Bonjour sleep proxy on the Airport Base Station). It's documented elsewhere that you need to have 'Wake for Network Access' switched on in Energy Saver prefs and your Airport or Ethernet hardware on the sleeping machine has to be capable of supporting the feature. The issue for me turned out to be that the AEBS is operating in Bridged Mode (I have an Airport Express further extending the network as well) and it does not transmit the magic packet in that mode. I purchased a Home Network plug and connected the ATV3 to that so it was now attached to the network via Ethernet and bypassing the AEBS. I turned Airport off on my iMac and it now works! It can take 30secs to a minute for the ATV3 to make the connection during which time it displays 'Loading iTunes Library' but eventually it establishes a connection.
When you look in the system log you will see 'kernel[0]: Ethernet [AppleBCM5701Ethernet]: Wake reason GIGE - magic packet received'. If, in your configuration you never see this message then the magic packet isn't getting to your iTunes machine for some reason and it will remain in sleep mode.
Hope this helps.