Unfortunately, I doubt that there is a body of good research, although you might try running your own searches on PubMed. Researching the effect of childbirth education is highly problematic by nature. Observational studies won't get you anywhere because classes vary in content and teaching style even among educators certified by the same organization. Moreover, couples who sign up for classes tend to self-select into classes according to their own viewpoints, which brings me to the problem with judging effectiveness by labor stats. For one thing, if a highly motivated couple seeks out a class with a teacher or method known to promote natural childbirth, would low epidural rates be attributable to the class or the prior intent of the couple? For another, the woman's care provider's and hospital's policies and practices will almost certainly overpower whatever the couple learns in their classes. Who, after all, are they going to believe: the trained experts or her educator, especially when they, as my Aussie friends call it, start "waving the shroud"? In fact, classes sponsored by a hospital or clinic where conventional obstetric management is the norm--and this would be the majority--would hardly encourage informed decision making. Any teacher who tried it wouldn't do it for long. A randomized controlled trial might tell us something useful, but RCTs are complex and expensive to mount. You would need an institution to back it, which circles back to the issue of the previous sentence, not to mention what would be the payoff for spending the time and money needed to carry it out?
In an ideal world, we wouldn't need childbirth education classes. Care providers would assist women in making informed choices by giving them complete, accurate, unbiased information about their options and providing them optimal care in labor (the least use of intervention for any given individual that produces the best outcomes). All women would labor under the guidance of caregivers and in environments that foster the natural unfolding of the process, including access to a wide range of comfort measures and pain coping strategies besides epidural analgesia. Ironically, the reason classes are needed is to help women cope with a system that is all too often "none of the above."
I'll step down off my soapbox now. Anyway, for a perspective on what role childbirth education could play in maternity reform, see this Science&Sensibility blog post.
-- Henci