Calculating average slope along line using ArcGIS Desktop?

A simple approach that you could automate in ArcMap using Python would be to break your line into points at a chosen interval or even just at the vertices. Then extract values to points from the raster to get the longitudinal profile (make sure to select the interpolation option). Then simply calculate the average value of those points.

As @whuber mentioned there is an accuracy trade-off with simplicity, but if you're looking for a simple solution that you could script quickly the above should help.