Some routers software that has not been updated my not work w Mountain Lion. This is the case w Linksys, I just found out yesterday when I was advised to update firmware. Harder to fix after the software no longer works.
But even w that, I'm still connecting, although it may be making the connection more intermittent.
Maybe some setting in preferences.
confused whether you are connecting w both cable & dsl.
But at any rate, have you tried resetting the modem & the router? I would think that would be a 1st step that I have taken when I didn't have connection. Also if you are on wifi, try connecting directly via ethernet to see if you have connection that way.
Usually to reset & I'm doing this by memory, disconnect both the modem & the routers power & ethernet. Supposedly just removing power for a few seconds should work but for my Linksys router (which I'm about to give up on & go get an Airport Express unless the emails I have waiting give me a quick solution) I have found I need to leave it powered down & disconnected for at least a few minutes.
& then if I recall correctly, you reconnect the modem & let it come up so the lights are all blinking. At that point, connect to the modem via ethernet & see if you have a connection. This helps pinpoint where your issue is. If you have a connection then proceed.
& then you reconnect the router, but have wifi turned off while it is doing this, don't have internet apps on or try to connect until it has fully come up for maybe a couple of minutes or so. & then go to the wifi icon & see if your network is selected (actually that should be a 1st step before all the other things. I've found that sometimes if I turn off wifi & then turn it back on, it will search for my network, find it & then be ok, so start w this).
If your router is an Apple Airport router, it should be simpler I think. The above is based on my experience w a Linksys router & older Motorola cable modem.
Sometimes for whatever reason the modem and/or router may get messed up & need this type of process. But again, it may be simpler hopefully w an Apple Router.
You might also want to go to the router website if non-Apple to check for software compatibility & updates for ML. But again, not sure that would cause totally loss of connectivity.
Hope this helps.