Alternative recursive DNS resolvers like Google's 8.8.8.8 (which gets the award for being the most-memorable) and OpenDNS, DNS Advantage, etc. are an option, but they are decidedly downstream of rootservers so they may have been affected.
There exists a non-ICANN-run collection of DNS nameservers [1] (based out of Europe) which are normally synchronized with the ICANN root, but can be operated independently.
If, for some reason, the rootservers become unavailable for an extended amount of time, others may move to establish rootservers, but every downstream DNS resolver will still be configured to depend on the rootservers [1]. Reconfiguring these relationships is a fundamentally human process (as opposed to a machine process), so it will occur out-of-band; then once trust is established the downstream servers will point to the rootservers' static IPs as it is currently done [2].
75.75.75.75
75.75.76.76