That's what happens when flattening transparency. Are the lines gone when you disable anti-aliasing of artwork on screen? Then everything should work fine.
You should read something on transparency. Never use it without knowing what you're doing.
I do not know if this will work for you but it might try bring the file into photoshop and save as a tiff or exporting it as a flattened tiff at least for the artwork itself without the text (if any) then place the tiff an Illustrator document and and then put the text in it.
See if that works.
It seems that you used some kind of blur effect in Illustrator (the glow under the picture, probably), and what you are seeing are the edges of the blur. Blur type effects in Illustrator are useful for making a first approach to an idea, and maybe for presentations, but not for final art or printing, they simply doesnt blur to transparent. Reproduce theese effects in photoshop using PSD with transparent background, for example.