Apparently the root of the problem is udev's handling of devices. Udev keeps a history of devices attached to the computer. What makes sense for e.g. USB sticks (that is, each USB stick you attach will be recorded and mounted to the same point when attached again at a later point in time), causes problems with network devices when systems are cloned.

There is a file in /etc/udev/rules.d/ - something like /etc/udev/rules.d/z25_persistent-net.rules, where udev records which hardware device should get named eth0. It looks something like this:

# PCI device 14xx:16xx (tg3)
ACTION=="add", SUBSYSTEM=="net", DRIVERS=="?*", SYSFS{address}=="00:11:xx:xx:xx:xx", NAME="eth0"

When you run the image on another computer, the new network adapter is recognised on the PCI bus and a new entry in the above mentioned file is generated. "eth0" is already given to device 00:11:xx:xx:xx:xx, so the new device gets named "eth1".

If you delete this file, it will be rebuilt during the next system startup and your problem should go away.

The file may not be called z25_persistent-net.rules, but the command:

sudo rm /etc/udev/rules.d/*

will get rid of it for you!