To get a glimpse of the future of commerce in America, look no further than Sweden.

The Scandinavian country is largely a cashless society, with consumers relying on mobile phone payments or plastic. While the U.S. is still far from achieving the same level of cash-free existence, increasing numbers of restaurants and retailers are now snubbing the lowly dollar bill.

Some merchants such as SweetGreen, a salad chain, refuse to open their registers for cash, telling customers they can pay only with mobile payments or cards. With some newer vending machines, only a card or mobile wallet will get that cold Coca-Cola to roll down the chute.

The stance may appear un-American -- after all, currency is considered legal tender for all debts or dues -- but the Treasury permits private businesses to set their own policies, which means going cashless is fine with Uncle Sam.

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"What we've seen is a push toward electric payments because of convenience, especially for Generations X and Y and onward," said Greg Burch, vice president of strategic initiatives as Ingenico Group, which makes payment systems for merchants. "The phone has become more personal than the wallet has."

While going cashless is still at what Burch said is an experimental level, he predicted that more stores will be adding mobile payment options or their own mobile wallets, similar to Starbucks' (SBUX) payment app. The coffee chain's mobile wallet is gaining in popularity: It now accounts for about one out of every five purchases.

Using cards or mobile apps is increasingly popular with younger generations, but stores have good reason to like the trend, as well. Moving away from cash removes the cost of storing and transporting bills and coins, which merchants like. It also reduces the potential for physical theft.

The downsides? One is a loss of anonymity because cash allows consumers to make transactions without a paper trail.

Another negative impact is stores that refuse cash may be effectively shutting out many lower-income customers. About one out of 13 U.S. households are unbanked, which means they have don't traditional banking accounts, such as checking or savings accounts. Such families tend to be lower-income and rely on cash to make their purchases.

While federal law allows merchants to set their own rules about which types of payments to accept, at least one state makes it illegal to refuse cash: Massachusetts. The issue came to a head recently after a rash of stores catering to well-heeled young professionals posted "no cash allowed" at their registers.

SweetGreen, the salad chain, changed its cashless ways in its Boston locations after learning about the rule, according to The Boston Globe. SweetGreen didn't return CBS MoneyWatch requests for comment.

"We've now adjusted plans in our five Boston test stores to be in compliance with Massachusetts retail law," SweetGreen President Karen Kelley told the publication "As we grow, we learn, and as we learn, we adjust: It's all part of our mission to do right by our customers and our employees."

Of course, the bigger hassle for consumers might be the flip side of going cashless. Many retailers and restaurants still have cash-only policies, which can prove increasingly irksome as consumers visit the ATM less frequently.

Cash isn't in any danger of disappearing, but maybe it should: The U.S. has much to gain by phasing out cash, according to researchers from Tufts University. Writing in the Harvard Business Review, they noted that the U.S. spends $200 billion each year to keep cash in circulation. (China also has high costs related to reliance on cash, they noted.)

They added: "Both the U.S. and China would do well to adopt policies in partnership with market actors to nudge their already digitally ready societies towards digital money and unlock massive savings -- in time and money -- in the process."