Part of what makes this work is our mutual knowledge that it would be very hard for either of us to game the system. I can't just print up new dollars and neither can you; this means that the dollars you give me are very likely to have been given to you by someone else, which makes it very likely that I will be able to pass them on to someone else, which makes them useful and therefore valuable. If I could just print up my own dollars out of thin air, I wouldn't actually need to do any work to acquire them, which would make me less likely to accept them in trade for whatever it is you want from me; if everyone else is in the same position, then none of us have any reason to accept dollars from each other, and the currency is no longer useful as a means for exchanging debts. So the usefulness of any currency depends on the degree to which it is difficult to create new currency units.
This is what I mean by confidence that bitcoins are going to remain valuable: people can't just pull them out of thin air. It takes a lot of work to get a bitcoin. Right now the bitcoin system is new so lots of people are mining bitcoins, but once there are enough of them it will be easier to get a bitcoin by doing some other kind of work than by mining. This is all we need for bitcoins to become a useful medium of exchange for IOUs, should we decide to use it that way.
What cryptography (reasonably) guarantees is that bitcoins aren't double-spent, aren't created from thin air. Not that they preserve value or are acceptable as payment.