As I walked to lunch it dawned on me that I might be able to get at
the attributes via the GetType() method.  Well, I wasn't able to, but
I did realize that as long as you follow the convention of <enum
name>TypeConverter, that through reflections you can instanciate the
enum's TypeConverter:

// start by using the default ToString method
string valueStr = Value.ToString();

try
{
    // Get the enum type
    Type enumType = Value.GetType();

    // Get the enum name and then add the TypeConverter to the end of it
    string typeConverterName = enumType.FullName + "TypeConverter";

    // Get the assembly that the enum is in.
    string assemblyName = enumType.Assembly.FullName;

    // try to create the type converter
    ObjectHandle objHandle =
Thread.GetDomain().CreateInstance(assemblyName, typeConverterName);

    // Get the TypeConverter from the object handle
    TypeConverter typeConverter = objHandle.Unwrap() as TypeConverter;

    // Convert it to a string
    valueStr = typeConverter.ConvertToString(Value);
}
catch(Exception)
{
}