Are there any modern crypto algorithms that are, by design, immune from an attack such as this? Would not having any key-dependent code paths be sufficient to prevent this attack? If it is possible to be immune by design to power analysis, timing and tempest attacks, is there a list of such algorithms somewhere that I can look it up? My google-fu hasn't returned anything useful.
The only 'provably secure' (e.g., on paper (+)) countermeasure you can apply to these symmetric schemes is something typically called masking. You can view masking as using secret sharing techniques to split up all intermediate computation into independent operations. To defeat masking an attacker needs to be able to re-combine the data dependent information leakage associated with all the split components. This is always a possibility.
Thus it becomes a risk/cost tradeoff. The more you mask, the more secure you become, but at a cost of speed/area/power draw.
(+) It's decidedly non-trivial to implement a masking scheme such that you get the theoretical security. This is an active research area.