Try this. It creates a DataObject and sets the DataFormats.Text datatype so it could be tested in the grid. It also specifies DragDropEffects.All so you do not have to press the shift to do the drop. Since the source is responsible for deleting the moved object, it will not do any harm to accept all actions in the grid.
private void Form1_MouseDown(object sender, MouseEventArgs e)
{
String s = "Testing...";
DataObject data = new DataObject(DataFormats.Text, s);
this.DoDragDrop(data, DragDropEffects.All);
}