I have had an account at ABN-AMRO in Holland for many years. It comes with their Wereldpas ATM card Maestro, good all over Europe. ABN-AMRO has a cash transfer plan in their Internet Bankieren program that works slicker than shineola. It is almost instant and no charge. You do need to wire your funds in at your home town bank rate, usuallly about $35 per transaction. After that you are a euro person and don't worry about the exchange rate.

If you are going over often (three times for me last year) I recommend it.

I used to have an account in Amsterdam, which I opened when I sold my boat there, and another in England for paying alimony.

I closed them a few years back when I was not going to Europe very often. Bad mistake! Wish I still had them.

I have read on this thread that it is really difficult open one now, so I have not tried. Any advise would be appreciated on how this is done for a non EU citizen.

Or even simpler, open an account with an American bank that has minimal charges for overseas use of an ATM card

If you need larger amounts of money quickly then open such 2 accounts

I have an account in France to pay for taxes and utilities, but I see no advantage to using their credit card, which is a debit card and would cost me a annual fee. I currently pay no annual fee and no fee for withdrawals with my ATM card, and one percent for using my Visa card on purchases, both of which have been accepted all over Europe. So what is the advantage of a card at a European bank?

It is not that easy to open a Dutch bank account unless you are a resident here. What Spaarne neglects to say is that he opened his account whilst living here.
Any bank account within the Euro zone offers free money transfers to any other Eurozone bank. It is one of the advantages of the Euro. Not every country has free bank accounts, such as we have here in the Netherlands. Indeed I think it is only a matter of time before they start charging us here too.

If you do decide to try and open an account choose another bank than ABN Amro. It is now owned by the government, along with the Dutch part of Fortis, but I wouldn't have an account with them, even before the current problems. I didn't like the service, or lack of it, from them. All Dutch banks offer internet banking, and all provide instant transfers.

Charles Schwab accounts don't charge fees for getting money from any ATM, worldwide. And if there is a charge from the bank (e.g., ABN-AMRO) in which you take out money from, Charles Schwab will refund you this fee.

Source: http://www.schwabbank.com/checking.do

hetismij has a good point. I went to the ABN web site and found the conditions for setting up an expat account to be rather onerous.