Are you sure you're in Continuous AF (not just continuous drive mode), where you've got a large focus spot in the viewfinder (the regular spot when locked, plus a circle around it)?
If so, how are your focus points set up (center, select, wide area)? I usually leave mine on select, and keep my thumb on the control pad. When I lock and release using a press of the pad, continuous seems to favor the selected point. Center is going to be fastest. So, you may want to try just Center.
You can also set the camera to shoot without a lock. I think the Canon models default that way shooting in AI Servo Mode (which is their version of AF-C ). That way, the camera is still going to try and take photos of where it thinks the subject is going to be, even without a good lock.
You see complaints from new Canon shooters from time to time that their first frame or two in a burst when a subject starts moving are out of focus. That's because when in AI-Servo mode, the camera is designed to shoot, lock or not, and it can take the camera a frame or two to get the focus right when movement starts (since it's predictive algorithms need to determine direction and speed of movement).
I never set mine that way, but you may want to set your 5D to behave the same way and see if that increases your number of keepers. Go to Priority Setup in the Custom menus and set it to Release versus AF, and it will shoot with or without a good lock.
You may also want to try AF-A mode. If you lock focus on a subject with a half press, it will automatically switch to Continuous when it detects subject movement.