Those stripes are due to not enough colors in your image. Either you didn't start out with enough, or somewhere in your processing you removed some of the colors. Without knowing exactly what you did with the processing, I can't really tell you exactly what caused them.
As for why they are so noticeable, that is called
"Mach Banding" and appears wherever you get two very similar colors together.
I can assume that you are using a color image, and them making it grey-scale. If you're using "posterize" in photoshop, you might try playing around with just the saturation.
Also, I would recommend you make changes brightness and contrast of the image before you go into "B&W" mode. By going into a B&W mode first, you have reduced your colors from some 16 million down to 256 immediately.
Good luck,
Brian