This is just a display issue related to the anti-aliasing algorithm most programs use to display vector graphics. The effect you see is not "real", and for most practical applications it is pointless to try to get rid of it. In printing and in exporting rasters these lines don't show.
To prove it, you can go into Edit > Preferences > General... in illustrator (Ctrl + K), and untick the "Anti-aliased Artwork" checkbox. The white lines will disappear. Other programs have similar options to turn off anti-aliasing. You just have to find them.
The downside to turning off anti-aliasing is that curved paths will now look less smooth. Some people might prefer this since it is less noticeable and certainly more logical than the white lines. Keep in mind that this is all still only display settings, and if you open the files in another program that has anti-aliasing on, the smooth curves and white lines will come back.
If you permanently want to lose sight of those lines you can use one of the many workarounds such as adding very thin strokes, offsetting the shapes to a very small distance, or making a copy below your artwork and scaling the group a little...
Use whatever solution works best for your needs.