HDSLR Movie Making Flickering Problem
There are a number of factors that could cause the effect you've described. It's unclear which you're talking about specifically, but here are the possible problems and their possible solutions:


Interlacing - Video filmed for broadcast is split into 2 interlaced fields. For NTSC, this means that instead of ~30 full frames per second, a camera will first film even lines for ~1/60th of a second, then the odd lines for ~1/60th of a second. Or for PAL, instead of 25 full frames, they'll do even lines for 1/50th of a second, then odd lines for 1/50th of a second. The solution to this problem is to shoot progressive (full) frames.
Rolling Shutter - DSLRs, cell phones, and other types of cameras use a CMOS sensor which scans the field of view from the upper left of the sensor horizontally, going down to the next row of sensor sites until it reaches the bottom left and starts over. This can cause artifacts when objects in the scene move faster than the scanning happens. (Or when the camera is moving.) The solution is to use a camera with a different type of sensor, such as a CCD sensor, or to apply a post process to fix the rolling shutter problem. Most video editing applications have a filter of some sort to correct this.
Shutter angle - When shooting video, you can tell the camera how long you want the shutter to be open for each frame. If you're filming at 25 frames per second, then it can be opened as long as 1/25th of a second. But you can reduce it to, for example, 1/50th of a second while still only shooting 25 frames per second. This may be expressed as either a time value (such as 1/50th of a second) or as an angle, where 360° is the full frame time, 180° is half the frame time, etc.


It's hard to tell from your description which problem you might be encountering, but you can try making changes for each. Canon cameras will allow you to film in either interlaced or progressive modes by simply changing a setting. Likewise with shutter angle. Fixing rolling shutter requires either a different camera or applying a post-process.

The Canon C300 has a CMOS sensor just like the 5D and the C100, so it's probably not the rolling-shutter issue.