Ask your bank that issues your ATM card the name of their corresponding bank in each country and try to use your ATM cards there to avoid surprise fees. If there are corresponding banks in each country you will need less emergency cash because there's no downside to withdrawing money as you need it. It's usually the best exchange rate you can get.
I still wouldn't go with any less than $1,000.00 emergency cash. As some have said here sometimes the US $ gets you a better deal.
btw, there are some great exchange rate app's available.