This is why Persia and Rome are a pain as neighbors. Great early UU and the resultant GA.
My advice, play nice, at any price. Trade, trade, trade give gold per turn whenever possible. With Persia I actually hope they will demand something from me because then I can give it to them and they wont attack me right away (immortals give me the begeezies)! With Rome I either wipe them out before they get their Legions or play nice.
The notable others (Zulu, Aztec, Iroq) aren't too bad, their bark is worse than their bite. As always bring in alies. This is another reason why trade is so important.