I'm a big fan of racing in general. I like the thrill of beating the clock, watching my back from someone trying to pass, the anxiety of pushing myself to the limits of trying to pass someone and all of that.
It's a really fun experience. However, there aren't many racing games and not very many non-traditional racing games for that matter. I have been thinking of a concept, and I'd just like to jot down a few of my ideas here, and gather some opinions and thoughts about racing games from you all as well. Feel free to contribute any kind of information or opinion as long as it regards the racing genre!
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No cars.
While it is still racing, there are plenty of other ways to race. By foot, board, bike, etc. There are already plenty of car racing games anyway. This also helps reinforce my next two points quite well.

[size="4"]No track.
How about a designated start point and a common location for an end point that is located on the other side of a map? This develops a hint of strategy because you have to find a good route from point A to point B. With a clever enough map design, there could be several routes, each with pro's and con's. For example, in an urban area, streets could have traffic that you have to race around. You could also jump building to building, but you'd have to spend a little extra time climbing or descending stairs/fire escapes/etc. But at least you don't have to worry about traffic.

[size="4"]Combat. Oh glorious combat.
Now this is where it gets even more tricky. Not only do you have to be aware of where other people are, you have to be aware of what they can do to hurt you and set you back. Couple this with a destructible environment and you can close off routes, knock people off of narrow passages, shift time, hamstring someone, etc. This could also be a nice test of reflex if there are such things as reflects and speed boosts.

[size="4"]Instead of customizable cars and car models, have unique player progression and spell choice.
This would introduce a concept of a metagame. You can only have a fixed number of slots to bring abilities that alter the map and players. It's up to you to pick between and mix and match to find something that suits your playstyle and strategy, as well as counter what your opponents bring. This would make sure that the same maps can be played multiple ways and keep the small rotation of available races to be unique most every time.

[size="4"]So what does this all give you?
Quite simply, a fast paced, off-the-rails racer with many intense moments and tons of depth for strategy, player development, teamplay, customizability, and fun.
This would develop into a game where you have to think about what you need to do while you're racing, before you're racing, while you're racing, and even afterward as well.
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What I would like to hear from YOU!
What do you think about what I said? Should I elaborate on something in specific? Would you like to hear more? What are your thoughts on the genre currently, and what ideas do you have for the genre. I'd love to hear from everyone, so post a reply!

You have great thoughts but I'm seeing games where these things have been thought of. Maybe not currently, but they have been expolored.

No cars: There are plenty of ATV games and BMX games that have some sort of racing involved. I don't remember a game that is strictly foot racing so that's a good idea but there are plenty of games that incorporate foot racing to complete a mission.

No track: This concept is done in Burnout Paradise and loosely in some Need for Speed games. Expanding on the "no cars" concept though, this leaves room for a lot of possibilities but again, this has been exploerd in various games but not the sole purpose of the game. The idea of racing with fire and other things as obstacles is a good one. Different on foot racing environments and depending on the skill level, you can do more acrobatic things to take faster routes.

The racing genre is already a pretty explored genre. From realistic to arcade, the genre has been tried in many different ways. I think foot racing is the most unexplored form of racing and with the technology today, it could be a lot more viable to experiment with.

There was a game called FUEL that tried to do an open world racing game. Unfortunately it didn't work out very well.

The actual idea I still think has merit, but it depends on how you handle it. I think the most important thing to remember is that when players can go anywhere, everywhere has to be interesting. That itself is a hell of a challenge.

I think the best example of this working is the grand theft auto games - they allow some stupidly fun open world racing with all sorts of vehicles in a world where all the routes are interesting. I think using something proven like that scaled down to it's most bare essentials could be very interesting. I heard a story on a podcast about GTA IV (not having played multiplayer myself) where there is a mode where everyone has to make it to a checkpoint on the other side of the city and all the players start on super high powered sports bikes. Right out of the gate most people go flying into the first car they see and start a huge wreck. The trick is every time you crash your bike gets downgraded, so as people make their way to the finish line their bikes are getting worse and worse each time. The podcaster said he was nearly dying with laughter when the final dash for the finish line came down to everyone on vespers desperately trying to knock each other off at 20 km/h