Like many others my Apple TV Gen 3 died after the latest firmware update (5.1), just thought i'd share my experience as it may save you time.
After firmware update to V 5.1 (5201) front LED blinks but ATV doesn't start.
Try to reset by holding menu and - on remote until light blinks fast.
I found that if i connected only the power and HDMI cables it would start up, but couldn't see any wifi networks. If i connected the ethernet cable (i don't use the wifi) it would restart, i would see the apple logo then the LED would just flash and it wouldn't start up.
Following a suggestion on this thread i connected the ATV to my computer using a micro USB cable, and plugged in power. Device could not be seen in itunes - tried another computer - same. Left it overnight- still nothing. Then i read a post stating that some micro usb cables are better than others, mine looked very cheap, so i borrowed one from a friend and ATV was now showing in itunes.
Did the auto restore in itunes and was back to having no visible wifi. This time it did start up with the ethernet cable connected, but it would not go past the language selection screen. I selected English, it would think about it, then go back to the language list.
So i followed the instruction to manually download the old firmware V5.0.2 and installed that via itunes.
IT NOW WORKS AGAIN!
2 lessons i learned, don't update the firmware unless you have 5 hours to spare - always use a DECENT micro usb cable to connect ATV to your computer.
Apple - this might be your hobby, spending hours fixing botched firmware updates is not mine 😟