As my router won't allow me to modify its DNS settings I ended up changing the configuration locally. The following set of instructions should work on OS X Lion. You might need to adapt the recipe for your purposes:
- Add your new DNS addresses to network settings. In short System Preferences → Network → DNS → DNS Servers (add 184.108.40.206 and 220.127.116.11 here for OpenDNS or 18.104.22.168 and 22.214.171.124 for Google DNS). Same thing graphically.
- Flush your DNS cache using sudo killall -HUP mDNSResponder at terminal. Other versions of OS X use a different set of commands.
- Flush your browser DNS cache. As I use Chrome I had to navigate to chrome://net-internals/#dns and hit "Clear host cache".
That did it for me. If you ever run into weird behavior with your networking, DNS might be a part of the reason. Apparently DNS provides some additional services so if you need something like parental controls, that might be the way to go.