Who provided the Cisco router?
I have 3 Cisco routers in use and have no problems connecting to any IRC network through them, once I got the various ports set up correctly using help from PortForward.com
I did have trouble with one Cisco router that was provided by my ISP, as it was configured by them for the purposes of providing a static IP address. It took awhile (they didn't want to admit that it was their equipment), but after about 2 months they agreed that the problem was with their router and replaced it with another Cisco router (different model), and I've had no further problems.
Thus if the Cisco router was provided by your ISP, and you can eliminate all of the other possibilities (ie: being able to connect with the router not inline, but not able with it inline), then it is your ISP's responsibility to provide working equipment.
If the router was not provided by your ISP, then I suggest you contact the retailer or Cisco for a replacement. Which you contact will depend on the warranty period provided by the retailer.