It depends on the lens and the situation. For the Safari shots I relied heavily on a monopod, only because I had one chance to get the shots I wanted. I was doing media work for the winning team and had to deliver for each day's press releases. It was a nuisance carting it everywhere though. For nearly all the shots I used a 70-200 F4.0L with image stabilisation (IS) but often turned the IS off because it doesn't always play nicely with a hard mount.
For track work I will often shoot with the same lens, IS on but off hand. The key to offhand work is to get your stance and support correct. It's very similar to shotgunning - support the camera with your leading hand under the lens, and put most of your body weight on your leading foor. With the panning shots, it pays to pick a bike 3 or 4 seconds before it ends up where you want to shoot. I take these kinds of panning shot with both eyes open. Once it's as close as the bike in the first shot I'm actually watching where it is with my left eye more than the right, almost sighting it over the lens. Stability comes if you swing from the knees and the hips rather than trying to track with your neck, mainly because the arms move less that way and I find it's my arms which introduce the really destructive wobble. Next time you're out at the track near a corner, just practice tracking the bikes without shooting. That way you start to get a feel for how much you need to move (it's a surprising amount) and you'll find the blackout time when the mirror is up less confusing. Then, pick your time, establish correct exposure and shoot in manual mode. If you can get down to 1/500 and still keep the bikes looking sharp you're getting the hang of it.
One of the super things with the 1D is the very short blackout time. It can rattle off 10 frames a second, but I normally knock it down to 5 or 6 unless I'm doing pass through shots with preselected focus. It does have some drawbacks though. The AF system is extremely complex and requires careful setup for each kind of shot. Set badly, and it will take pictures as bad as a ten year old pawn shop dunger. It took me a long time to come to grips with it and I still get caught sometimes.
I've used other bodies in the past but there's a reason that you see lots of these and Nikon D3/D3s bodies out there mounted on big lenses. Once you know what you are doing they make it much easier. But you can LEARN what you need to learn with anything.

