Luckily, credit cards are really a second tier payment system in most of the EU. Country-specific (debit) cards and the international Maestro payment system are the primary way to pay, along with cash. Apart from online shopping, of course. But even there, you usually have other viable payment options. So it's probably not all that much chaos...
Many online shops take SEPA direct debit, which may be superseded by SEPA instant payments in the near future.
Basically the same flow as paying with paypal but completely through my own bank