But I guess it still remains to be seen if lots of modern Rinzai teachers are going down the Brad Warner/Stephen Bachelor path of making lots of noise about not believing in traditional Buddhist ideas about karma and rebirth.
To start, there aren't a lot of modern Rinzai teachers anywhere (meaning those who were named as successors and lineage holders by their teachers, i.e. they received inka shomei
The ones I've known, both Japanese and western, did not negate it at all. Rinzai Zen is Buddhadharma.
But one assumes that rebirth teachings, like any other, are to be employed in appropriate circumstances. Answers given by various teachers might well have been appropriate for the listeners present then: a person fixated on a concept of substantial self might find the carpet pulled out from under him/her ("When you die you'll disappear completely"), while someone fearful and lacking confidence regarding their ability to practice might be encouraged not to worry because "if you don't get it in this life, you'll continue in the next existence". These are two answers I witnessed coming from the same teacher.
Regarding the "post-mortem rebirth" vs. "rebirth within moment-to-moment existence" debate, it is taught clearly that the the latter does not negate the former, but is rather just another skillful means to be applied when appropriate. For example: These future results become again the next cause, and the cause again produces results: a circling around without end, from birth to birth revolving through endless Kalpas. This is the turning on the Wheel of coming to be and ceasing to be. But since it is difficult for us to comprehend events so far in the future, we can look at it in a more immediate interpretation and consider it as the principle of birth and death from moment to moment. In our own hearts thoughts arise and thoughts die away again; we inevitably have to accept this fact.
[Daibi of Unkan, teisho/commentary on Torei's Shumon Mujintoron
So short answer: Rinzai Zen is Buddhadharma, but the focus of Rinzai practice is what it considers the essence of Buddhadharma, the recognition and integration of one's nature - which is not bound by Buddhadharma or any other teaching. Resolving that, nothing remains unresolved...and so teachings of rebirth or any other teaching will be (hopefully) employed skillfully to support the student's practice. A Rinzai teacher's job is to bring about that recognition, and guide the student along the path of integration which follows. As far as the means used or set aside, nothing is a sacred cow and ultimately there is no fixed method.
Of course I describe the ideal situation, in which a teacher is deeply realized and the student has sufficient capacity. I am not saying every teacher is deeply realized. We know that not every student is suited to this approach. Research well to choose your teacher and path wisely.
Finally, another possible source of confusion is that the classical model of Rinzai practice stresses that it is crucial to examine and grasp the purport of the sutras, but typically not until an advanced stage of practice, and then primarily for the purpose of checking one's experience to see that it tallies with them. Of course in the past, students coming to a Zen teacher would already have had some training in basic Buddhist teachings.
But in the west today folks from all kinds of backgrounds, mostly Abrahamic in terms of early religious training, come walking in. Added to this is the fact that a large percentage of them, having been so well educated on "what Buddhism is" by way of popular books and online forums, are fairly certain they already "get it".
The end result: lots of self-defined Zennies with a small amount of actual Zen experience who think something like, "Zen rejects traditional Buddhist teachings and ignores rebirth. When I went to the Rinzai place, we didn't read anything. They just taught me a basic practice and told me to do it for a couple of hours a day as a foundation. When I asked about rebirth and karma they just told me to keep counting my breaths".
At our place here we have a regular study group examining basic texts and teachings, to try and fill in this background which new students often lack.