Profanity in ancient times largely revolved around bodily functions and sexual acts (fun fact: the word “motherfucker” actually comes from ancient Greek). Comparisons to animals were also popular (such as “female dog face,” which slowly morphed into the modern word “bitch”). And of course, there were still religious curses. One could swear by Zeus, Hera, Apollo, or whichever deity your city was dedicated to. And if you didn’t have the stomach for full on curses, there were still euphemisms. Those who didn’t want to swear by a deity could shout “By the dog!” And if you didn’t want to tell someone to eat shit and die, you could tell them to throw themselves to the crows.
I don’t know nearly as much about swearing in Latin, but I found an article that contains some information about ancient Roman profanity. It looks like bodily functions and sexual acts were again the main source of vulgarity at that time. You might also be interested in the book that the article is about, which chronicles the history of swearing.
@YARNLADY “Zounds!” is a shortening of “God’s wounds!” and is a reference to Christ’s suffering on the cross. It definitely does not predate Jesus.