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There are regular travel rewards credit cards issued by major banks that will allow you to redeem miles with any airline, including with Emirates. You will find miles rewards credit card offers from Discover and Barclays. Such cards are not tied to one particular airline. In fact, they are not tied to any airline and miles you will earn are basically cash back rewards redeemable for travel and flights in particular. And since Emirates do not offer airline credit card to the US consumers, a travel rewards credit card can be the best option for you.
Take a look at the Discover it® Miles, for instance. It's a perfect option both for avid travelers and regular people. You get 1.5 miles on all purchases, which are matched at the end of the first year (provided you are a new cardmember). That's basically as much as 3 miles per dollar, it's more than some premium cards give you on purchases and more than all airlines give on airline purchases. Simply, that's a lot.
Discover is applicable to all airlines and not only then - there are numerous opportunities to redeem your miles. What's more, with Discover you are sure you don't overpay - there's no annual fee ever, there's also no transactions fee, which allows you to save your money while traveling - a lot of the cards have a transactions fee, which is not what you want to pay on vacation.