Can I work directly for a company abroad (Sweden) in the United States if they do not have a U.S. entity and I'm a U.S. citizen? Or should I set up an LLC and function as a consulting business?
It's very easy. No need for an LLC. I'm in a similar situation. The only thing I've asked of my (Irish) employer is that they issue me with a 1099-MISC form in time for tax season.
It will mean you'll have to sort our your taxes yourself (pay estimated taxes and pay an additional payroll tax) as well as deal with benefits (e.g. health insurance).
But it works fine for me, and has for over 7 years.
I recommend finding a good accountant to help you with it.