essentially both sides were doing this by tapping wires into telegraph lines that they saw. The basic work around was to use codes which at this point in time were not very well developed. if I recall early in the war some units were still sending messages in "clear" (uncoded) but this obviously changed as the war went on. Most codes were not very complex, again going from memory but I think the best codes at this point were digraphic substitution codes with a key (a series of letters that changes the value of all the characters) that was not very long.

On the other hand, I dont think they were that easy to break simply because there werent thousands upon thousands of messages sent in the same code so you didnt have the luxury to logically go through all the possibilities. They might grab one message here and another message there and then a day later the code has been change.