I think FDR removed that custom altogether. If the 22nd Amendment was not enacted, his successors would have felt perfectly enabled to run for as many terms as they wanted. In practice, however, the issue would have only affected 4 presidents: Eisenhower, Reagan, Clinton, and Bush II. Personally, I believe the first three would be under strong pressure by their party to seek a third term as they were still popular among key constituencies, and all would have probably won. Eisenhower and Reagan might have declined to run again because of health and age issues, but Clinton almost certainly would have - and won the election. It's hard to imagine Bush II running for a 3rd term, but as an Obama supporter, I sure wish he did.
As an aside, I think the 22nd amendment is one of the worst ideas.