If your Linux systems are running a normal Linux distribution, go update your kernel. And then keep updating them over the next few weeks. If your distro does not have kernel updates, then I strongly suggest changing distros right now.

Meltdown – x86
Linus’s kernel tree contains all of the fixes we currently know about to handle the Meltdown vulnerability for the x86 architecture. Go enable the CONFIG_PAGE_TABLE_ISOLATION kernel build option and rebuild.

Spectre
the status is there is no fixes merged into any upstream tree for these types of issues yet...Because of all of this, it is going to take us in the kernel community a few weeks to resolve these issues and get them merged upstream. The fixes are coming in to various subsystems all over the kernel, and will be collected and released in the stable kernel updates as they are merged...