This is not very compatible with linux at all.

Everithig is OK

Wonderful card. It is inexpensive and works without any trouble. I have over 5 of them and i love them!

I had a problem no exactly with the NIC (network interface card) but with the APIC (Advanced Programmable Interrupt Controller). This is a mode that made this card not working on Mandriva 10 and Debian 3.1.

I fixed this in Debian by editing /boot/grub/menu.lst. I added "noapic acpi=off".

Example:
title Debian GNU/Linux, kernel 2.4.27-2-686
root (hd0,0)
kernel /boot/vmlinuz-2.4.27-2-686 root=/dev/hda1 ro noapic acpi=off
initrd /boot/initrd.img-2.4.27-2-686
savedefault
boot

In Mandriva I used the boot section of the control center. There is a check box for APIC mode.

r1000 was only supported by Ubuntu from the distros I tried (mentioned above).

(By the way, thought windows says RTL8139/R810x, it is most likely not an RTL8139, or it would use the RTL8139 specific driver, and linux detects that well.)

I downloaded the latest source for the R1000 drivers from Realtek, which required some editing before getting to work, here is how:

These are the steps I followed for r1000 v1.04

I copied the r1000 files into /root/r1000
such that I had /root/r1000/src/..... files
and /root/r1000/Makefile

I edited /root/src/Makefile_linux26x
and changed
include $(PWD)/src/Makefile_linux26x
to
include /root/r1000/src/Makefile_linux26x

then I edited /root/r1000/src/r1000_n.c (thanks to dotato)
and changed the 3 lines startting with MODULE_PARM to MODULE_PARM_DESC

as root I wrote
make clean modules
make install
depmod -a (no idea what it does but it was in the readme)

then I ran r1000.
Started up SuSE's YaST, went to 'view hardware'
Closed it.
Opened up Netword Devices (still in YaST) and there was my NIC ready for configuration!!!

Joy oh joy!
It finally works.

Once you get the driver installed, it works well, Don't know why r1000 was not included with any of those Distros except Ubuntu.

I don't have this chipset, but have helped a number of people who do.
If you have ever run windows on your PC you need to read this thread for reliable operation:
http://www.linuxquestions.org/questions/ubuntu-63/weird-network-card-problem-591466/#post2922729
The easiest solution is at post #11, and it seems that windows can shut the chipset down, so linux is unable to see it.
For the purposes of this feedback form I've rated it as "Excellent" as when it works, it does so well.