I go for the diplomatic way usually, so if someone is stupid enough to go to war with me, next turn they usually find that i've bribed all their other neighbors into declaring war on them.
The one time I got to play, I was the Romans. During the course of my game I only discovered the Greeks and the Egyptians on my map. At first I tried to be diplomatic, and with Alexander it worked fine. But Cleopatra just wouldn't budge on any deals I proposed to her, including offering writing for pottery. So I got frustrated and killed her.
I will try to have more patience in the real game, because shortly after that Alexander got a little bit more wary about me. Although he did agree to some pretty bogus deals out of fear…
Certain civs are a lot harder to deal with than others, especially militaristic ones. Civs like the Aztecs and Zulu will rarely bargain with you, and if they think you have so much as a city or colony that they can easily take, they'll eagerly declare war on you with no warning. If you're on the same continent as one of them (the AI absolutely sucks at naval warfare, or for that matter exploration), you need a strong defense no matter how diplomatic you are.