It does not seem necessary for The One to undergo death and resurrection because all indications are that Neo has always been The One. The One is a designed function and carries Machine code which is used to reinsert the prime program, so the Machines play a role in determining who is The One.
The only requirement on the part of the human potential to function as The One is that he must be convinced that he is The One. As the Oracle put it: "Know Thyself". The Oracle told Neo that he would have to sacrifice himself for Morpheus so that Neo would convince himself that he was The One, but the sacrifice itself was not what made Neo The One. Neo's belief that he could rescue Morpheus from a military-controlled building that was also guarded by Agents was an indication that he was beginning to believe -- Morpheus said so himself after Neo saved Trinity from the helicopter (which was before his resurrection).
Trinity's kiss and revelation to Neo that the Oracle told her she would love The One constituted further confirmation to Neo that he was The One and made him believe it more strongly, but he was already The One by that point.
Neo's predecessors therefore only needed to be convinced that they were The One by any means -- whether that meant a death/resurrection, rescuing someone against all odds, etc.