This is definitely a constant problem that I find there is no one simple solution for and, to make it more difficult, there usually is no way to know until you try to print and see it happening. You say you exported a PDF from InDesign, does that mean you are printing straight from a PDF after it was exported from InDesign or you are printing from InDesign? This is usually a transparency problem that occurs with Illustrator so I'm not quite sure what to tell you. In Illustrator I will usually try flattening transparency, if that doesn't work then I will try Rasterizeing the artwork but most times the file becomes too large for Illustrator to rasterize. When that happens, saving as a PDF and then printing that will sometimes work but if not....I will import the file into Photoshop and create a flattened image there that I will then import back to Illustrator to print. It kind of depends what the artwork is and how the problem is occurring, sometimes you can manipulate only the part you are having a problem with and not need to do the whole file. I find it to be a trial and error process to find a solution and it's always a different thing that works from the last time. I hope someone else can give you a better answer.....
