Accepting cards from most locations isn't actually difficult any longer with Stripe, Braintree, Paymill, and others. US cards aren't special in terms of being able to accept them; they're just regular Visa/MC/Amex/Discover/whatever cards.
The neglected payment methods are the country specific ones. Some major countries have very low credit-card penetration, for example in the Netherlands. The common internet payment method there is iDEAL which is a kind of two factor authenticated bank transfer. Even debit cards in the Netherlands have two distinguishing features: there isn't a PAN number on the card which makes it impossible[0] to use online in the way you'd use an embossed Visa/MasterCard, and the Dutch banks all issue their debit cards under the Maestro and V-Pay schemes.
[0] Not really, but very few people know what their debit card's PAN number is. It's possible to calculate the hidden PAN number as it's usually just a regular Maestro card, or to just scan it off the card using NFC and an appropriate app.