it depends on what you mean by: "anonymous Visa cards". There is no such thing. There are debit cards, credit cards and atm cards. Even ATM cards (Visa Electron cards) will require that you provide documentation on who is the owner of that card. Parex will require a pile of documents from you. each card has to be assigned to a person (even if they are issued to companies). your passport, signature and life story has to be on file with the bank. If the cards are issued to a company, you need to include a description of your business and explanation of why you need the cards, when they will be used, how often, how many deposits, how many withdrawals etc... even if your name is not on the card - same is true for each card holder or for each card issued even if they are issued to a company.
dont think that any country entering the EU is going to protect your privacy in any way.