Over the years I have met very few Western Buddhists including Zen Buddhists who fully accept the idea of rebirth, that consciousness (vijñāna) does not die but is, in fact, born again in a new embodied condition based upon certain karmic propensities (this idea being contained in the 12-nidānas).
Also, I found that in discussions with many Western Buddhists they don’t believe in first-person out of body experience reports or first-person near death experience reports even when the evidence is quite good, including results of the late Dr. Ian Stevenson's reincarnation research of children who reported remembering a past life. Their position is quite skeptical and not without a deep-seated, unconscious bias that privileges the mechanistic or Newtonian world view which rests on reductionism, determinism, reflection-correspondence view of knowledge (that knowledge is just a reflection of objects outside of us) and materialism.
The idea that, according to the Buddha, our consciousness embodies itself and upon the death of the organism finds a new embodied condition is quite unbelievable for most people. We prefer, "I came from a blank state and will return to it. There is no life after death." But Buddhism also wants us to know that this consciousness that transmigrates is an attribute or vibration of mind which is the substance of reality—what our imaginal reality is made from.
It is almost mission impossible to try and move the Newtonian cult out of the university where it has been around for a very long time. We still like the dark mechanistic world. Despite that somewhat bleak outlook I do see more and more evidence that the Newtonian, clockwork world is giving way to a break in the von Neumann chain. The break in the chain, where all physical devices are links in the chain including even our brain, has to occur when finally matter gives way to the mystery of the extra-physical mind which is not a link. This mind is who we are. It is also deathless. It does survive physical death. By realizing it we come to the end of blind, uncontrolled rebirth.