The anthropologist Donald Brown came up with a list of human universals some years ago in an attempt to disprove cultural relativism; the list contains dozens of entries, all of which are considered present in every single human society. (For political reasons I know some people are reluctant to believe in an innate 'human nature'; if you are one of those people, bear with me.) There are too many to reproduce here, but they range from "affection expressed and felt" to "males engage in more coalitional violence" to "childhood fear of loud noises" to "incest, prevention and avoidance".
My idea is this: people often complain that demihumans are, like Star Trek aliens, usually just human beings but with rubber heads. There are often vague notions like "dwarves don't like the sea" and "elves are good with magic" but nothing much more interesting or detailed than that. So why not use the list of human universals? Pick a universal, change it to its exact opposite, and try to extrapolate from there to create something markedly different from us.
Here's an example: dwarves have no concept of 'hope'. Their assessments are always brutally honest. If there is a good chance of succeeding at a given task, they'll attempt it, but otherwise, they won't - and cannot be persuaded. This obviously makes their societies extremely conservative and risk-averse, and the speed of change glacial; when a dwarven society sets itself a goal it is only ever one that can be realised through tiny, incremental steps that are relatively sure to be achieved.
Elves do not trade and have great difficulty understanding the concept. If an elf has something that you want, the only hope of getting it is to take it by force, or to try to persuade him that you need it more. If you can, he will likely give it up and expect nothing in return - the idea that goods should be exchanged being completely alien to him. This likewise extends into the more abstract realm of exchange of services. Diplomacy between humans and elves is fraught with difficulty because so many of the crutches of human diplomacy - trade agreements, tit for tat, quid pro quo - are incomprehensible to elves. Persuasion and cajoling are the only viable negotiation techniques.
Halflings have no abstraction of speech or thought - everything relates to the physical world or emotional states. Halflings can't be asked to imagine situations that they haven't encountered before, and find 'if/what' propositions impossible to comprehend. Conversely, they are highly sensitive to physical reality and almost empathic in their ability to understand the emotions of others.
For gnomes, the biological mother and social mother are different people. When a gnome woman gives birth, her child is given to a trusted friend to raise, who will likely reciprocate when she in turn has a baby. The giving and recieving of children is an important bonding mechanism in gnomish society.
And so on. It works for 'evil' races too: perhaps orc societies don't proscribe murder, or goblins have no sense of what a promise is.
Of course, changing some universals is pretty difficult to envisage. I challenge anyone to explain how a society would function in which there was no such thing as 'choice'.