The Czech page on obtaining the EU vaccination certificate based on a non-EU vaccination mentions that only Czech and EU citizens are eligible for now. So i.e. if you’re an American tourist (without dual citizenship) traveling to Prague with your CDC vaccine card, you would not be eligible for an EU vaccine certificate. This will make travel a bit inconvenient as you’ll be forced to get tested frequently, as this is the only official alternative if you don’t have a valid certificate of vaccination and want to visit things like restaurants and museums.
But are there countries which are more open minded and will issue the certificate to anyone who applies? Ideally without ever having to travel there physically? This way you could obtain the EU vaccine passport remotely from (say) Estonia and then use it in every other EU country during your travels.