How can I straighten ALL the vertical UV lines at once? Is there a hot keyshort cut? Or if I have to do them one by one .. is there at shortcut for that? Thank you!
The UVSquares addon is designed to do this easily. Select the uv's you want to straighten out and click a button.
There is no shortcut for that unfortunatelly.
In Uv Editor select the corner of the UV island, press Shift+S-->Cursor to Selected. Change the pivot type to 2D Cursor. Select the bottom edge of your island and press S,Y,0 to straighten it. Now press P to pin the selected vertices. Do the same with upper, left and right edges. Next, go to the 3D View window and unwrap your mesh again.
Now straighten the other edge loops using Shift+R command. Change the pivot type to Median Point, select the edge and scale it as described above. Then select another edge and press Shift+R to repeat the scaling process.
Not always applicable, but worth trying if your UV map is already nearly gridlike: Make a very tiny temporary image, with as many pixels as your mesh has columns, and enable snap to pixels, then select all your verts and drag them around. There will probably be some hand cleanup afterwards, though.
In Blender 2.8, you can also do it with the built-in addon MagicUV.
Here is a simple geometry I created.
The red lines are the marked seams. In the UV editor, you can see the original UV was not straighten.
All you need to do is select an end edge and press N > in Magic UV Tab > in UV Manipulation check Align UV > check Transmission > hit Straighten or XY-axis.
And the line will be straighten.