Most effiecient and inexpensive way to travel in Europe

'It looks like the best way to fly between the cities would be by air.' Well, yes, you got that part right. :) Moving on, smartly: Open jaw is the same as multi-city. Most people use it for US-London-Rome-US - - something like that. It's better than a pocketfull of one-ways. Your trick is to find something similar (if it exists with a single carrier) that would get you from Amsterdam to Turkey, Oslo?, Barcelona, and some place in Germany, thence back to Amsterdam. If you've found a way to pull that one off for six hundred bucks you might have found the pot of gold. If you haven't, then we'll assume that you can hop a train from Germany to Amsterdam easily enough and work from there. So it's now Amsterdam-Turkey-Sweden-Barcelona-Germany. Damn, it's sure help if you gave us the missing cities! Plug this into kayak.com using the multi-city function (it's right by 'one-way' and 'round trip' and see if anything falls out. If it doesn't, then you're down to booking individual segments on individual budget carriers - - which may not use the same airports as the larger carriers, so watch out that you're not stuck umpty-seven miles out of the town you're trying to get to. You can also try using kayak for mult-city for only some of your points and fill in the gap with a budget carrier. Come back if your fingers wear out. Or give us the cities and somebody might know exactly how to do it.

You didn't give cities. You might get very lucky with EasyJet, depending on your cities. They fly in and out of several cities in Turkey. Ryan Air is also a budget airline, I've avoided it because I've heard that the service is dreadful. But it's cheap, if it covers your cities (check the airports they fly into, it isn't always the big one...). Good luck with the flights. You're planning to cover a lot of territory. Some night trains might work out for you too.

No offense, but your goal of efficient and inexpensive does not jive with traveling to the four corners of Europe and in/out of a city not even on your list. So as long as you understand that your travel expenses will be high, there is nothing wrong with hitting the places you want to visit. Other than Amsterdam/Germany (depending on city), air will be the only realistic option. Check the discount airlines like Easyjet, Veuling, and Ryan Air, but many mainstreet European regional airlines like SAS, Olympia and Iberia can offer very good prices. Keep in mind that baggage restrictions are very tight and strictly enforced, so expect to pay extra fees. Some of these airlines will fly in/out of remote airports. Still, for the time and money saved versus trains for those distances, air will be the best value. You're just going to have to try a lot of combinations and compare the costs.

Why don't you take the train to Sweden then fly Turkish Airlines open jaw from Sweden-Istanbul-Germany. It's a very good airline.

www.skyscanner.com for no-frills airlines. Also www.whichbudget.com although a recent makeover of the site had the dazzling result of making it even clunkier to use.

Amsterdam is not the best hub for intra-European travel as flights from and to it are expensive (due to a lack of low cost airlines serving the city). To get to Turkey take the train to Düsseldorf (DUS) or Cologne (CGN) and book Air Berlin, Pegasus, Condor,... flights to turkey. Search for flights using Kayak's German division, Swoodoo or Momondo. Book a roundtrip from Germany to Turkey as that's cheaper than a one-way. Then from the same airport (or a nearby one) book a cheap roundtrip to Sweden. Roundtrips between Germany and Sweden are likely to be cheapest on Ryanair, but both Lufthansa and SAS offer €99 roundtrips.