Grain is something inherent when one shoot with 35mm, or 16mm or Supre 16mm... And most directors want the grain because that's one of the beauties of true 'analog' celluloid film.

Most now have software which handle digital videos sort of effects to add digital grain to the film to make it more film-like...some looks ok, some not - depending on the process

personally - dont really like films without grains...esp those shot with too high end digital cameras - mainly because they look too clean and too digital...i think those hi-end HD cams like the Cine Alta or Genesis could actually shoot without adding this so-called noise or grain into the picture - but normally the director / editor / producers would insist in putting a little bit of synthetic grain into the video to give it a more film look/

but on this note, in the digital domain - im not sure if we could always equate grain with noise...because i believe they are quite different.

so yeah - i guess digital isn't always a good thing...it looses the 'magic' of traditional film materials...