I feel the same way you do about multiple bikes each year. I end up not doing a thing.....I guess that is waht feels most ethical to me. I totally disagree with the leave a note, wait a week, cut it off approach.
Is a new old bike really worth trading in your honor? Even if it is abandoned, their is no way you could ever be sure. And that is what makes cutting it off dishonorable, imo.
On a related note, my buddy and I longed for a steel lemond alp-de-huez with Campy Veloce left outside at our work over the winter. I was almost certain it had been abandoned.
It changed positions again when it warmed up. Turns out it belonged to the Security Guard!!!!!