It varies a lot depending on who you ask, and most postmodernists probably don't really have a theory of truth. Some really don't believe it's a meaningful concept, or even if they do, think it's epistemologically never accessible, so not worth bothering with.
Some are open to coherence theories of truth, though, more or less that you can't decide whether something is "really" true, but you can decide if sets of beliefs are coherent with each other, within a particular culture's conception of coherence. That's actually not too different from coherence theories of truth that a lot of logicians hold, which is that "truth" is consistency relative to a formal system. Whether that turns into anything-goes seems to depend on just how arbitrary the choice of formal system is.
They really like to take it all the way, don’t they? They take two undeniably true (ahem) statements like “absolute certainty and hence absolute truth don’t exist” and “culture influences truth finding processes and the status quo within cultures makes changes in certain directions easier or harder” and carry it to excess. They turn mere problems with how humans can generate true statements into central pillars of their doctrine.
What’s the big deal? There are certain statements which allow us to make useful predictions. (I would call them true statements but I won’t insist on that.) We have to be vary of our biases, we have to be ever vigilant in our search for useful (true?) statements and it’s a damn hard process. But it’s not impossible.
On that I definitely agree. I think in terms of their interests and biases, most people in the broad "postmodernist" camp want to find any variety of certainty impossible, whereas I'd come from the perspective of agreeing that there are a whole lot of problems and complications with certainty, but I actively want to work towards it rather than being happy that there are so many bumps in the road.
Some are open to coherence theories of truth, though, more or less that you can't decide whether something is "really" true, but you can decide if sets of beliefs are coherent with each other, within a particular culture's conception of coherence. That's actually not too different from coherence theories of truth that a lot of logicians hold, which is that "truth" is consistency relative to a formal system. Whether that turns into anything-goes seems to depend on just how arbitrary the choice of formal system is.
A different direction is Richard Rorty trying to take some branch of postmodernism and weld it with early-20th-c. American pragmatism: http://en.wikipedia.org/wiki/Philosophy_and_the_Mirror_of_Na...