Remove it. If it is soft, you can do it by hand, otherwise use a stump grinder. Replace it with topsoil. The wood, if left in the soil, will rob it from nitrogen, and also sink in once decomposition takes place. I always try to be thorough, because if you don't, you will end up with sunken patches and possibly non uniform lawn coloring from different nitrogen levels in different areas. You can add the root material you remove to the compost pile, as it contains valuable organic matter, you can take advantage of.
Refer to this answer for details on filling in, and this one for details on removing the stump/roots. They don't entirely pertain to this situation, but they should be helpful as guidelines for the removal and cleanup of stumps/roots. If you are regrading, you might as well fill in when you do the rest of the work.
You could also try burning it out, but this isn't the best idea as it kills the soil organisms and doesn't improve soil health. Composting in place (by piling high nitrogen matter on the stump/roots until they decompose) is a healthy method environment-wise, but has the potential to take years.
