If your Chromecast isn't rooted, it all depends on your router.
There are 3 main methods:
The most widely-supported method is to get the Chromecast to fall back to DHCP by using the static-route-to-nowhere method, since almost all routers support setting up static routes. You can read about that method in the thread you referenced above.
A few routers let you block the Google DNS addresses specifically to make the Chromecast fall back to DHCP. That works just as well. But unfortunately most routers that follow the Linksys model of access restrictions can't block a specific IP address, even though it looks like it should be possible the way the interface is set up.
If you have a router running DD-WRT or other advanced Linux-based firmware, you can use the iptables command to redirect the Google DNS addresses to the DNS addresses of your choice, as in the link you referenced.