Although Sachleen's answer is preferable in a lot of situations, there are a few other options that you have.
Quick note on his answer: I find clamping a row of those wires together and rubbing a thin line of superglue over the connectors a useful way to keep them together. It's not very neat, but it works pretty well and, with that many wires connected, can make it more stable than before.
There are two main alternatives I see:
Ribbon cable: very similar to the wires pointed out in Sachleen's answer, only the connectors come already as one piece.
Source: http://www.adafruit.com/blog/wp-content/uploads/2012/06/window-207.jpg
They are typically two rows, but I've seen them before with one (although pretty rare). You could just ignore one row of those and use some male headers to connect the same side of the ribbon cable to the breadboard/Due.
DIY: although you said that you didn't like soldering, it's still the best solution here IMHO. You'll need soldering later on, and breadboards aren't reliable beyond just simple development, so you'll need to learn at one point or another. This is a great entry project because it's pretty hard to mess a simple thing like this up, unless you accidentally bridge two joints. However, you won't damage an expensive chip if you mess up (which you will on the first few times you solder), you can just clean the joints and start over.
Anyway, just grab some perf board and some wire (twisted pair (similar to CAT5e)/ribbon cable with or without an adapter and plug/loose wires, and a few rows of male headers. The type of wire you choose doesn't really matter, just don't get something too thick or too thin. Anyway, solder some headers to the perf board and then you can solder the wires/the socket (if you choose a ribbon cable with ends) together. You can then cut the perf board to size (you can do this step before soldering) and you're done!
