The text of the text input box is automatically assigned if SelectedItem is set initially. This should happen when the SelectionBoxesVisibility property is set to Visible (which is the default value). For example, if you have an item called "Item A" and you assign it to the SelectedItem property at startup of the control, then the text will also become "Item A".
To work this around, you can manually clear the text on loaded of the control.
private void RadMultiColumnComboBox_Loaded(object sender, RoutedEventArgs e)
{
var mccb = (RadMultiColumnComboBox)sender;
var tb = mccb.FindChildByType<TextBox>();
tb.Text = string.Empty;
}