No, that's not the only way. But the editor will look at the OLD link -- if that works, and if there's nothing on the site to suggest it should be changed, then it won't be changed.
(This is important for people to understand. Suggestions are just that, suggestions. We don't know who's making them, or why -- and we don't want to have to care. Reality is the website. Websites are what we review, and where we look for information.)
The notice could be as simple as a "we've moved!" link or a meta redirect...or all the links could be absolute links to the new site. We aren't particular, we just have to see SOME kind of evidence on the old site.
The 301 redirection is, of course, best, because that's the official standard way of telling EVERYBODY on the net you've moved. And webmasters who think of "ODP editors" as somehow different from "every other surfer" have fallen into a delusive snare.
So don't worry about the ODP. It will follow along with everybody else -- when you get the word out to everybody.
But at the ODP, there are about four levels of priority (which editors don't have to honor, but very often do--in fact, we asked for them, because we WANTED to honor them.) And in the highest priority is "automatically detected 301 interrupts." Update requests is #2 or #3. Ordinary site suggestions are last.