I'd like to note that "the Internet" is a vast, broadly-scoped amalgamation of routers and different network topologies. They don't use one kind of hardware or software to manage it all. Any successor or parallel alternative network should be as (if not more) flexible to achieve it's goals.
I'd also like to suggest that the network be powered purely by standard Internet client machines and off-the-shelf hardware. Custom software would be necessary, but it's better to rely on a random guy with a quick installer on a USB key than custom hardware mesh routers deployed by professional installers.
Lucky for us, the new IEEE 802.11 was ratified in September and includes (finally) 802.11s for mesh networking. There is support in Linux and FreeBSD for several of the most popular wireless drivers. The open source router firmware dd-wrt also supports 11s.
That is pretty damn cool, but I think for a serious "alternate Internet" to succeed, OSI layers 1 & 2 should not matter. An application installed by a regular user needs to be able to do most of the heavy lifting with most generic off-the-shelf hardware to get a really decentralized open alternative to take off like wildfire.
I'd also like to suggest that the network be powered purely by standard Internet client machines and off-the-shelf hardware. Custom software would be necessary, but it's better to rely on a random guy with a quick installer on a USB key than custom hardware mesh routers deployed by professional installers.