I feel like this is a common misconception amongst some physicists. Lorentz invariance doesn't "appear naturally" in modern theories depending on how you develop it. Usually, one chooses a lagrangian that yields a lorentz invariant action and so all physical laws and thus solutions are lorentz invariant consequently. Lorentz invariance is a fundamental assumption...upheld by experiment. Those theories will then admit solutions (usually linearized ones (read quantized)) that appear as waves.