How safe is it to use a Linux live CD to access my savings wallet?
Security is always an issue and complex systems like operating systems are prone to attacks. However, it is possible to use an offline machine to create the signed transactions and then export them to another machine. 

For example a soft copy of those signed transactions could be copied on to a fresh USB stick, or printed out on paper as plain text or QR codes and so on. These transactions could then be imported into an online Bitcoin client that does not have the private keys, but can just transmit the transactions to the network. 

Your private keys are always held offline and thus only exposed to physical attack vectors which should be sufficient to relieve your concerns.