As usual, figured this out as I was posting, but I'll post the solution anyway in case it helps someone.
As each column is removed, the index for each column changes. So when you remove column 3, 4 becomes the new 3. This code works:
for (int x = 0; x < csv_datatable.Columns.Count; x++)
{
csv_datatable.Columns.RemoveAt(3);
}
This will loop over the number of columns, and remove the third column over and over again until everything is gone.