"Postrationalism" is our reaction against some of the silliness of modern conceptions of "rationality", while keeping the strong emphasis on correct thinking, skeptical evidence-based inquiry, and field performance as the final test of ways of thinking. Postrationalists tend to be concerned with matters of social psychology and how societies work more than traditional rationalists. Let's explore some of the issues postrationalism addresses:
Propositional belief is too narrow of a concept to model all desirable cognitive content. It is better to think in terms of a general continuum of forms of cognitive content: memories, models, heuristics, skills, procedures, habits, and such, with truth as a sometimes-applicable proxy for usefulness rather than an always-applicable end in itself.
Ideas about what's important and valuable are usually thought of as fundamental truths or moral axioms, but are often better modelled as shifting social fictions, local residue of larger-than-human social phenomena, tribal markers, and so on.
Some rationalists have a reductionistic and mechanistic theory of mind. They see the mind made up of a patchwork of domain-specific biased heuristic algorithms which can be individually outsmarted and hacked for "debiasing". While the mind is ultimately a reducible machine, it is complex, poorly understood, very clever, and designed to work as a purposeful whole. You generally can't outsmart your mind. It is therefore better to treat the mind as a holistic and teleological black box system, and deal with it on its own terms; experience, intuitively understandable evidence, good ideas and arguments, and actual incentives. The mind is already well-tuned by evolution, and can only become wiser with lots of specific knowledge and experience, rather than more rational with a few high-impact cognitive hacks.
We can't really replace common sense and intuition as the basis of reasoning. Attempts to virtualize more "correct" principles of reasoning from math and cognitive science in explicit deliberative reasoning are unrealistic folly. We can learn useful metaphors from theory, and use mathematical tools, but theory cannot be the ultimate foundation of our cognition; practical reasoning is either based on reasonable common sense, or bogus.
Additionally, postrationalists have an appreciation for tradition, ritual, modes of experience beyond detached skepticism, "non-rational" sociopsychological phenomena, and other things traditionally rejected by skeptics and rationalists.
Some further reading on the subject: