What is the cheapest way to get to Europe from the USA?
It's usually cheapest to fly, though if you have the time you may find a deal on a repositioning cruise in the Spring. Many cruise lines have ships that sale the Caribbean in winter and the Mediterranean in summer, so they reposition the ships in Spring and Fall. Find March 2017 to June 2017 10-14 Day Cruises to Transatlantic lists some of the ones coming up this April. (It's a bit late in the year to book these, many bargain fares may be gone, but you never know.)
It is also possible to travel as a passenger on some cargo ships. - Cargo Ship Travel | Passenger Ships | Freighter Cruises | Container Ships
In terms of finding a cheap flight, I would check sites like Orbitz - Vacations, Cheap Flights, Airline Tickets & Airfares where you can compare flights from different airlines. If you are flexible about your destination an travel dates you'll have more luck. I just tried a few quick searches for round trip flights from Denver and found a flight under $800 to Dublin and one for $515 to London. The best rate I found to Berlin was more than twice that at $1,422. But when I did a separate search from London to Berlin (also round trip) I found fares under $200.
Fares change frequently, but if you plan ahead, have flexible plans and do your research you can find some decent rates.
For a great way to travel Europe see here: Eurostar: Still the best hack for economy travel in Europe.
Set up alerts for price changes at online sites like Google Flights. The cheapest flights in terms of day of the week and connecting airports and destination probably change quite a bit.
Make sure you check for fights to multiple European destinations.