In addition to adding the default gateway, I suspect that your Linksys router will need to learn of the 192.168.0.x network as well, otherwise when it receives a packet destined for 192.168.0.x, it won't know what to do with it (since it'll have no routing path to it) and it'll just send the packet to the default gateway (typically the WAN interface).
Most SOHO routers have the option for Advanced Routing, where you can add the route to 192.168.0.x.
The other option is to enable RIP on the Cisco router and the Linksys router. I actually did a similar experiment like this yesterday, and after troubleshooting for a while, I realized that the issue was with the lack of a route to the 192.168.0.x network on my Linksys router as well.