While not exactly rituals, Thucydides describes the intense superstition of the Spartan generals. They would wait for a sign from the gods before making any large military or political decision and would even refuse to fight, or postpone battles if they saw what they interpreted as unfavorable signs.

In fact, during the (I believe) Sicilian campaign of the Peloponesian War the two sides, Athenian and Spartan, delayed battle for several days after an earthquake.

Overall though, Thucydides describes this behavior as strange and even worth of ridicule, suggesting that it was not common for other Greeks to subscribe to this level of superstition.

Regarding the Romans, Machiavelli talks about these rituals in his Discourses on Livy. He mainly talks about the use of Auguries, or individuals who would ritually sacrifice chickens and attempt to read signs in their entrails. Machiavelli dislikes this practice and ridicules these men, calling them chicken-men. His also tells the story of one Roman general who would ask for the Auguries opinions, but if they were not to his liking, he would either put them in the first rank, as essentially cannon fodder, or throw them off cliffs. This suggests that the practice of Auguries, or chicken-gut-tea-leaves, was common but that the underlying superstition was something that wasn't always present.