I do not own a Canon 1D Mark III or the Nikon D3, therefore my analysis below is just speculation, and should be taken with a pinch of salt.
Like everyone else, I have read the reports of AF problems with the 1D Mark III. The question is what could have caused these problems and how likely it is that the problems have been resolved in 1D Mark IV.
The AF sensor design changed between Mark II and Mark III. The design of the Mark II design is shown below, followed by the sensor design in Mark III (extract from Canon white paper).
Note that the cross-hair sensors are all clustered together in Mark II, whereas in Mark III, they are interspersed with line sensors. Now look at the Mark IV layout (from Canon white paper):
Although the Mark IV has more cross-type sensors, the layout in automatic mode is the same as that of Mark III. In manual mode, the layout allows more cross-type sensors to be used. The horizontal line sensors are usable with f2.8 and above, whereas the vertical line sensors are usable with f5.6 or above, except for the center AF point, which operates as cross-type upto f4, and as a line sensor upto f8.
Comparatively speaking, all of Nikon's cross and line sensors are fully operational at f5.6 and above.
My take is that the Mark IV may not provide as significant an improvement over Mark III as it could have. I think Canon should have clustered all the cross-type sensors in the middle, just as the Mark II did, and as Nikon D3 does. Another potential issue is the lack of vertical sensitivity below f2.8 in any except the central sensor.
Apart from the AF sensor and the AF algorithm, the other factor that affects AF performance is the mirror operation. The Mark IV specification of the mirror hasn't changed from Mark III. Blackout time is still 80ms, and shutter lag is still 55ms (with 40ms only at max aperture). It seems that the shutter lag is limited by the speed with which the aperture can be closed. Canon has maintained the top speed of 10 fps in the Mark IV.
The Nikon D3 has a shutter lag of 37ms and a blackout time of 74ms. Despite this, the D3's maximum firing rate is limited to 9fps, which means that the D3 has more time in between shots, both for the mirror bounce to settle, as well as for AF calculations. Of course, we do not know how good Canon's mechanism is for reducing mirror bounce; the better this is the more time will be available for AF operation. It may well be that Canon's active mirror control mechanism, first introduced in the Canon EOS 1V, is more efficient than Nikon's.
For an interesting insight into Nikon's implementation, have a look at this video (credit: the video was posted at dpreview.com by user Tokyo24 on 28 Nov 2009):