So Map Algebra (see: A quick tour of using Map Algebra—ArcGIS Help | ArcGIS Desktop ) con be used to mimic the behavior of the Cut/fill tool.
If you have a DEM that represents the situation without the dirt sediment and another one that includes the sediment, calculating the difference between the two (and mask them for the ditch area) can be used to sum the data and calculate the volume. This will require enough your DEMs to have sufficient precision in order to get a good estimate.
What is the spatial resolution of you DEM? How wide are the ditches? Can the "rough construction measurements" be used to create a 3D representation of the ditch without sediment? Is your DEM (with sediment) based on recent data (situation might have changed if data is not recent)?
Can you share a part of your data to see what you have and if it is possible to get the results you're after?