I went to a very conservative seminary and read Girard's Scapegoat, though the overall position of the school disagreed with it. However, that we read it indicates its prevalence in academic institutions. Others I know who went to mainline seminaries, however, are very well versed in Girard.
In Rowan Williams' (former archbishop of Canterbury) stunning book, Resurrection, he writes in the forward to the revised edition, "At the time of writing, I had no knowledge of the writings of Rene Girard on the scapegoat theme; and it was not for a few years after the publication of the book that Girard's ideas began to circulate widely... The first chapter in particular needs to be read alongside Girard..." Given Williams' position as the then-head of the Anglican communion, I would say Girard is pointing toward something important.
This is tricky and depends on where you are coming from. Conservatives will probably disagree. Liberals and moderates may often embrace it. (I can't back that up, though) But I'll point you toward some starting points. First, it strikes at the heart of your working understanding of Soteriology.
(1) Penal Substitution - Crudely defined in legal terms: A crime was committed, a judgement was rendered, Jesus became the saving substitute in the verdict. Adherents to this form of Soteriology are more conservative and will have nothing in common with Girard.
(2) Christus Victor - Jesus conquered sin, or death, or even the Devil with his work on the cross. This view doesn't directly address Girards claims. The beauty of Christus Victor is that it can often compliment other views. Therefore, many who identify very strongly with Girard's work may also hold some form of Christus Victor.
(3) Moral Transformation - Jesus was an example of how we ought to live. This definitely has tracking throughout the Church's history, but its more recent proponents have been fairly liberal. I would argue that proponents of this view feel very at home with Girard. In fact, the weaknesses of the view (apolitical, denies importance of death, etc.) are bolstered with the scapegoat theme.