1. Dig out your old telephone modem and learn how to use it again.
2. Recruit ham radio operators.
3. Buy a satellite phone that has a built-in modem.
Even easier to shut down than the internet. Any communication which requires a third party medium is risky. After Egypt's actions I'm looking into ham radio equipment.
Long range it would be interesting to get wireless, self organizing network equipment which would act much more like the internet did originally when it was primarily seen as a decentralized communication method meant to survive a nuclear attack rather than a more efficient, but more centralized, network it has become. I had suggested that when Iran shut down their internet access when the Iranian government lost the voting but won the counting. I didn't expect that we might be the ones needing it.
Of course the network equipment's firmware would have to be open source to verify that no secret cutoff switch was built in.
Ham radio bulletin board software permits a variety of access mechanisms: HF, VHF, microwave, internet, dial up, etc. Good for communications amongst a small group.
The Winlink system allows you to connect by radio to foreign internet servers for email exchanges.
None of the above lets you “browse the net” as you can normally. I guess you could use dial up to an unaffected server but that would be slow and who knows what will happen to phone lines.
A printed copy of this list would probably be a good place to start.