I have a Asus M4A785D-M PRO MB with a on-board Realtek RTL8111/8168B NIC.
Unfortunately, the r8169 module shipped with the kernel does not work with that card.

To make it works you need to install the r8168 module and blacklist the r8169 module.

To blacklist the r8169 module, add the following line at the end of your /etc/modprobe.d/blacklist.conf file:
blacklist r8169

I made a package of the latest version (8.018.00) of the r8168 module,
you can downloading from the link below and install it like this:
$ sudo rpm -ihv dkms-r8168-8.018.00-1.noarch.rpm

This package use dkms, it's easier to manage and upgrade kernel modules with it,
after installing the rpm you can check if the module is there, then load it:
$ dkms status
r8168, 8.018.00-1, 2.6.33.6-147.2.4.fc13.x86_64, x86_64: installed
r8168, 8.018.00-1, 2.6.32.11-99.fc12.x86_64, x86_64: installed

if the module is not loaded, load it and restart the network service:
$ sudo modprobe r8168
$ sudo service network restart

There's also the src.rpm in attachment and you can check my rpm github repo for the spec file and the patches.