Different languages have different sounds that flow easily in that language. Names from another language are bound to be slightly mis-pronounced, especially if the new language doesn't have the original sounds easily available.
A great example of this is Chinese (Mandarin), which has its set of syllables, and isn't built to handle new ones. When I was in China, my name was rendered as Kaimei. Most native Chinese speakers simply couldn't get their head round an R after the ka sound. This in addition to the troubles with Rs and Ls. In this case, you could say that the Chinese version of Carmi is Kaimei, but I wouldn't say that I took on a Chinese name.
In a similar fashion, if an ancient Hebrew man called Yokhanan had to do business with the Greeks or Romans, they would end up calling him Yohan, which is a set of sounds more familiar to their ears. In the same way, a Roman called Yohan (probably Iohan in latin) were to try doing business in England, he'd end up a Jon. This is exactly we we have John the Baptist, whose Hebrew name is originally Yokhanan.
My point is that people didn't necessarily convert their names, they just ended up being called something that sounded familiar to the speakers of the host language.