In an ideal world, you wouldn’t have to use EQ when mixing music. Not only would all the sounds be perfect, but they wouldn’t overlap with any of the other sounds. Of course, this is not a perfect world and at the very least, you’ll just about always need some EQ (removing energy at certain frequencies is typically better than adding) to help two sounds that have lots of overlap in the same space on the frequency spectrum (piano, guitar and vocals for instance), come through the mix. Other ways to make this happen include panning and of course, track volume. Usually the best mixes use all of these techniques and more.
Here is an article by Björgvin Benediktsson that has some tips on how to mix guitar and vocals without using EQ: