Alternative to the visitor pattern?

You might want to have a look at the Strategy pattern. This still gives you a separation of concerns while still being able to add new functionality without having to change each class in your hierarchy.

class AbstractShape
{
    IXmlWriter _xmlWriter = null;
    IShapeDrawer _shapeDrawer = null;

    public AbstractShape(IXmlWriter xmlWriter, 
                IShapeDrawer drawer)
    {
        _xmlWriter = xmlWriter;
        _shapeDrawer = drawer;
    }

    //...
    public void WriteToXml(IStream stream)
    {
        _xmlWriter.Write(this, stream);

    }

    public void Draw()
    {
        _drawer.Draw(this);
    }

    // any operation could easily be injected and executed 
    // on this object at run-time
    public void Execute(IGeneralStrategy generalOperation)
    {
        generalOperation.Execute(this);
    }
}

More information is in this related discussion:

Should an object write itself out to a file, or should another object act on it to perform I/O?