Do you just want to check if there is a network connection, or if the network actually goes out to the internet.
In the first case, then do as it says in the "How do I check for an network connection question". This will let you know if the cable gets pulled out of the back of your PC, but its won't tell you if your modem or phone line is malfunctioning (for example)
The second case is more complicated, as far as I know, there is no way to check this apart from constantly polling some IP address. In the simple case, you could just have a Timer ping some website every 30 seconds in a separate thread, and raise events when it decides the internet is down. Be aware that ICMP pings are not 100% reliable though, occasionally they will get dropped even if your connection is ok.
If you really wanted to get complicated you could create a windows service raising these events, then multiple applications could hook in and consume the information, but thats a bit beyond my knowlage of windows programming though.