I would agree with bar tape on flat bars. I've done this and it is much more comfortable than any rubber grips. Remove them first though.
Do you feel your hands painful as if you are putting too much weight on them? It may be worthwhile to raise the bars by about 1/2" to see if you notice any improvements.
If your bars are genuinely straight MTB types, you will probably find they give you sore hands regardless. Add some bar ends for improved position or change to a different style of bar.