To answer your question: it depends.
As previous posters have said, you don't need a particular race for a fantasy setting. It depends on what role the race plays in the game. If the setting is markedly different from standard D&D, you generally either have to change the race or not use it.
For my part, I went a different route. I didn't take the race as a given and develop different cultures, although I have done that before and enjoyed it. Instead, I thought very hard about what I enjoyed about each race and what they mean to me and the role I generally prefer them to play. From there, I restructured some things. While the revisions can encompass the stereotypical aspects, it is possible to go beyond them and still remain true to the basic idea. These are the basics of what I have:
Humans
In most settings, humans are the dominant race, but few explain why this is such a common thing. Most would say that it's laziness, but I believe it goes deeper than that. Humans are usually the force of familiarity. We can relate to human beings because we are human beings. This familiarity helps to make the unfamiliar all the more exotic. In many cases, humans also embody the spirit of adventure. While not necessarily present in individual humans, this is true on a collective level. Look at the knowledge human beings have gained and the things human beings have done. What traits have allowed us to do this? We learn, adapt, and apply the knowledge and skills to other things.
Elves
When I think about of elves, there is something that is absolutely essential for me to see them as something other than skinny humans with pointed ears: magic. Elves do not simply use magic; they live in magic. Some would even argue that they are magic. It infuses every fiber of their being and is imbued in everything they make. Their magical natures grants them some immunity to the ravages of time. Age afflicts them only through magical means, and death only comes to them from sickness, accident, or violence. It also increases their perceptive abilities. As creatures of magic, they are more attuned to forces beyond ordinary sensory capacity, and this attunement enhances not only their physical senses, but their psychic senses as well.
Ironically, elves share much with human beings, but the influence of magic renders them almost alien. Their perceptive abilities make them more sensitive and refined than other humanoids, but the things they respond to with these abilities are not easily communicated to non-elves. Their virtual immortality makes them simultaneously more detached from and more involved with the life's joys, hardships, and changes. Elves can exercise remarkable patience when making plans or enduring hardship, using time (or rather, their own timelessness) to their advantage. Yet, this knowledge that all things pass with time makes them pursue the fleeting, mortal things that interest them with a sense of urgency that would be surprising even in humans. When motivated by love or hate, they can be relentless. Since death is an anomaly as opposed to an ever-present reality for elves, they can be disturbingly blase or intense about it.
Orcs
These replace the PHB half-orcs.
The most striking difference between orcs and humans lies in how they view themselves as individuals, or rather, how they don't do so. While orcs are certainly aware of themselves, their self-concept is not an abstract entity, but is intimately tied into their environment. Oftentimes, who they are is what they are doing at any particular moment. As Omar Orc nibbles on bread, he does not think of himself as "Omar Orc eating the bread," but simply "eating the bread" or "bread tastes good." On the one hand, this gives them tremendous focus. It is not easy to distract orcs. However, this single-mindedness also makes it harder to learn new things, particularly abstract concepts, which leads to the second major difference between orcs and other humanoids.
Humans, as a general rule, are reasoning animals. They perceive and understand through logic, with instinct being at best secondary. Orcs, however, are primarily instinctive, with reason taking a back seat to gut feelings and intuitions about things that cannot be understood rationally. Neither orcs nor humans lack reason or instincts, but each shows a preference for relying on one or the other. This difference in orientation has led some to consider orcs as nothing more than animals and thus not sentient, justifying all kinds of abuses against orcs which would never be tolerated in humans.
Gnomes
Note: I have changed these a lot, so bear with me.
Gnomes are similar to elves in that they are essentially magical humans. However, unlike elves, who live in magic, gnomes literally and figuratively make magic. Many myths feature artifacts which have mystical properties or where someone creates something that has powers greater than its exterior indicates. This extends not just to tools and weapons, but aesthetic things as well. The power of an image, of words, of music and dance, even of food, is undeniable.
Gnomes typically stand under 5 feet (150 cm) tall when fully grown. For this, they are often called dwarves. They hate this. The way gnomes look is a study in extremes. There is nothing of the merely average about them. Their hair, for instance, comes in only four colors: stark white, golden blond, jet black, or flaming red. It is either silky smooth or coarse as wool. The eyes are always very clear and strongly blue, gray, green, or black. Skin complexion is either ruddy, pallid, or dark. Their physiques are either stocky or wiry, but they never seem frail or weak-looking. Gnomish hands are large and solid or slender and elegant. This extends even to how gnomes present themselves. They are either unkempt, even unhygienic, or meticulously groomed.
Gnomes define themselves through their craft, which is any endeavor that requires dedication and discipline to achieve competence or mastery. For better or worse, gnomes are what they do or make. Even as infants and toddlers, they display an urge to create. As they grow older, they are irresistably drawn to their chosen crafts and all things related to it. Most often, this urge is expressed through a prodigious talent for the fine arts, performing arts, and crafts. Conservative gnomes tend to focus on perfecting traditional models and methods while those who are more interested in novelty pursue innovation in their chosen fields. Regardless of their orientation, they have a perfectionist streak ten miles wide, though how this affects their personalities run in extremes. Some are the pillars of patience and stablity, while others are ill-tempered madmen. Individual gnomes can often express both tendencies.
Halflings
"Halfling" is basically a template that can be added to any race that is significantly shorter than the standard.
