Eager for truly transparent boxes

Question

On several graphics for a book I'm writing, I need to draw thick, transparent grayscale lines atop existing graphics.  I've drawn them with skinny, long boxes because lines, while transparent, come out too light.  The intention is to make them look like I'm using a highlighter on the graphics.  (I'm using ID5, btw.)
Anyway, I've selected no stroke at all for these very thin "boxes", and given them a transparency setting of 50%.  While the transparency is working out great, most of the boxes also show anything from a medium to dark stroke along the right-hand side of the box only.  This doesn't show on the screen, only in print.
I tried adding a stroke to the outside of some of the boxes, but no improvement.
Worse, when I print the same page more than once, boxes that didn't have strokes on the right-hand side, get them, even though I haven't moved anything around on the page!  So there is really no rhyme or reason to what's going on
On a related note, some of the graphics I've imported from another program to Illustrator (for a color change) and then to InDesign produce a dark stroke along the right edge of the bounding box.  I'd like to prevent these also.
What to do to prevent the dark strokes?  Thanks!

Answer 1

Thank you, lilia, for getting me on track.  I had never heard of overprinting before and needed a tutorial.  As I didn’t want to download the image at the link you supplied, I went to Adobe help and from there navigated to
http://help.adobe.com/en_US/indesign/cs/using/WSa285fff53dea4f8617383751001ea8cb3f-7038a.h tml
where I could read about what overprinting is.  So I draw three circles, overlapped them and filled them, and then went through all the overprinting business to see how it worked.  Then I went to work on the situations in my book, and it worked!
The only tweak I would make to lilia’s good instructions is to go to Window\>>Output<<\Attributes to get the overprinting dialog box.  It took a while to find that on my own in the Window menu.  Viewing the overprint is easy to do.  The only possibility for the music graphics is Nonprinting, and if applied, the music will not print!  Good to play with that, too, to see what it does, even if I don’t completely understand why all of this works.
The funny thing is that what I did behaves more like >under<printing, because I had to remove the Transparency setting and then bring the music graphic forward of all the highlighter boxes I’d drawn.  I also backed off on the grayscale to only 8%.  All the dark edges on boxes are gone now.  Doing the happy dance here!
I hope this helps others out there.  I come to the forums to get answers and am grateful to lilia for responding quickly and kindly.  Lilia’s heart for training comes through, and I thank you.