Thank you, JP Karlsen!
I ended up paying with my gf's card, but I had to use the solution to be able to use my card online. 3D Secure is the only way to deal with this from now on.
Another info: My banker told me that some companies might not have updated their payment system, but this will most likely be very small ones. Therefore, you might not be able to by some places, but I am pretty sure Origin should have updated their system.
For the Danish users finding it a little difficult I'll write the solution in Danish below:
Online handel
Grundet EU-regler skal kortet igennem Nets sikres igennem 3D Secure, Denne side hjælper dig med at tilmelde dit kort til 3D Secure og få lavet en kode:
https://www.nets.eu/dk-da/digital-sikkerhed/Pages/Verified-by-Visa-og-MasterCard-Secure-Code.aspx
Denne procedure skal åbenbart følges for at handle online, men den påvirker ikke køb i fysiske butikker (fandt jeg ud af efter en snak med min bank).
Nogle netbutikker har endnu ikke opdateret deres betalingssystem ifølge min bank - blot en lille ekstra viden, hvis der stadig hersker problemer. :-)