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Cyan comments on A Fervent Defense of Frequentist Statistics - Less Wrong
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The biggest Bayesian objection to so-called "classical" statistics -- p-values and confidence intervals, not the online-learning stuff with non-probabilistic guarantees -- is that they provide the correct answer to the wrong question. For example, confidence intervals are defined as random intervals with certain properties under the sampling distribution. These properties are "pre-data" guarantees; the confidence interval procedure offers no guarantees to the one specific interval one calculates from the actual realized data one observes.
I'm personally pretty comfortable with such "pre-data" guarantees as long as they're sufficiently high probability (e.g. if they hold with probability 99.9999%, I'm not too concerned that I might be unlucky for this specific interval). But I'm not necessarily that interested in defending p-values. I don't dislike them, and I think they can be quite useful in some situations, but they're not the best thing ever.
I do, however, think that concentration bounds are really good, which I would consider to be a sort of conceptual descendant of p-values (but I have no idea if that's actually how they were developed historically).