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Once I moved to France I realized that local banks protect their turf BIG time. I could not open the account in Orange phone company's bank because it would not accept payment/deposit from an American or Luxembourg's credit cards. The company's rep used her own (french) card to help me! Credit cards are on an average 150€/year and some online banks give you a credit for purchases, so at the end, it may not cost you anything. I did not like the setup and continue using my American credit cards that are not only free,  but earn air miles. I opened an account with Capital One that does not charge foreign transaction fees and earns 1.25 miles for each $ you spend.
In France I opened the account with Credit Agricole,  which is not as cheap as online banks, but at least it is reliable. It charges me 4.5€/month (unheard of in US), but that was an acceptable option to pay your bills and write checks.
In general, I found that French banks are not customer oriented and not as sophisticated in their technology as US banks. Wiring takes at least 3 days, transactions are not shown online for about a day and getting a hold of someone un the bank is limited to their business hours, which even exclude lunch time. Forget about calling at night or even after 5 pm.
I will most likely have to open a credit card account with a French bank because I cannot even pay for parking with my American card,  and my card was refused several times at the Government related offices. Well, c'est la vie:)

I was surprised to see that cheques are still in very common use in France.  I still use my debit cards the most, but I also find that I use cash a lot more than I did in New Zealand because in provincial France there are still a lot of places that don't have card facilities like some taxis for example.  I do have online banking too and a french banking app.