The onboard LAN can be disabled in the BIOS, Advanced, South Bridge Configuration (not that a Z77 board has a "South Bridge", it's a PCH, but whatever...), check if that is the case.
I use a separate NIC card, with the Onboard Lan disabled. With no cable connected to the onboard LAN, none of the LEDs are on. I have a simple multi-port network switch, so I can connect multiple PCs to one Internet connection. I unplugged the network cable from a shutdown PC, the other end connected to the network switch, and plugged it into the onboard LAN jack. The LEDs came on quickly.
You did not mention what your network cable is connected to besides the PC, so I'm going on blind. Try another network cable. Verify that the network connection is working on another PC. Some users have had problems with the ASRock XFast LAN Utility, do you have it installed? It usually auto-starts with its display showing. It is not required to have this program installed for the network connection to work, I personally don't use it. Try uninstalling it.
No idea what OS you are using. The disk with your board may have older drivers, have you tried the latest one from your board's download page?
ASRock > Fatal1ty Z77 Performance