You will need to look at the routing table. Use the command 'route' (may require root privilege) and it will show you the path that it wants to take to the other machine, including the interface. What you will want to see is that traffic for 192.168.0.0 with an appropriate netmask, e.g. 255.255.255.0 is routed out the third NIC card. This means that all traffic for machines on that network segment will be handled by that interface.
Since you can't ping, it is possible that this may be the issue. On the other end, or rather on both ends, you will also need to make sure that you don't have filters in place on the firewall for that interface. Since it is a private interface, you may want to set it to accept all traffic on that interface.
Lastly, make sure that you have a running process on the 'other' machine that is listening for traffic. You may then need to look at the configuration for the process and make sure that it isn't bound to listen only on one of the other interfaces. This will be likely in the configuration files in the /etc drive.
I would start with SSH as this is a pretty easy one to tell if it is working or not. In its configuration, erase or comment out and interface binding. You can also code 'netstat -nta | grep 22' to see if it is listening on port 22 and this will also show you the IP that it is bound to. If it says 0.0.0.0, it indicates that it is listening on all addresses (interfaces).
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