I found it very instructive to see how the $E$-$k$ relationship of a free particle can be roughly identified from the extended band structure of a solid: The following is the outcome of the one dimensional Kronig–Penney model.
For three dimensions things get more complicated. Lets have a look on Si:
Taking only the 100-Direction and putting the slices together, I come to this "extended" zone scheme:
The red and blue branches are what I expect from the one-dimensional picture. But, to put an example, where is the yellow branch coming from? It doesn't fit in to the "simple" picture.
I always thought, that that there is a unique k(E) Relationship for each Brillouin-Zone and the red, blue and green branches are from first, second and third Zone, respectively.
Now I see, that my picture cannot be completely true, because the yellow branch doesn't fit into my model in mind. Can the yellow branch assigned to a particular zone or is my idea of a one-to-one relationship Branch<-->Zone not reasonable at all?