This creates the moral dillemma of whether to be good for the sake of a future self that you will not remember and for the world in general as opposed to the concrete, selfish goal of an immediate, material reward (like the promise of heaven)
The idea of NOT remembering removes the carrot... in the most stripped-down forms of Buddhism, you are good because you have decided to be good, not because you want some fabulous shiny reward of virgins, angels, wealth, eternal life or to be off the hook for your sins.
It's sort of like the Egyptian notion of weighing the heart against a feather. When it's all over, your deeds are measured. If you have done more good than bad: you move up in the system. If you have done more bad than good: you move down. Only "you" isn't "you" at all... just the seeds of your soul.
It's very much like having children in a way. You die... but you continue to influence the world through the direct product of your own deeds. Your next incarnation will carry your karma, and your influence, into the future.
In essence, Buddhists must face the reality that their consciousness will end, quite possibly forever, when they die (like atheists). But they have the added burden of considering the progression of their karma and the implications that their position in the cycle will have on the world and the system as a whole...
Buddhism functions as both a philosophy and religion. Belief in the afterlife or reincarnation isn't necesarily required to believe the philosophy... I'm not sure how "religiously" Buddhist I am. But in essence: whether I believe the afterlife tenants or not ~ the outcome is the same.
dust to dust...
Reincarnation wasn't necesarily meant to be a way to "explain away" death or make it easier to accept. It isn't an easy answer... most people purporting belief in it misunderstand how potentially heavy the concept really is.