White balance is just the first step.
1) Buy a good neutral grey tone card and use it. They are only about $15-$20. You can't count on other items to actually be neutral.
You take a photo of that card in the same light that you are shooting in, with the card filling the frame, then you set the "Custom White Balance" using that picture.
Using Auto White Balance will give you a series of images all with slightly different color balance. If you first take a picture of people in front of a bright yellow wall, then in front of a blue wall, the camera will "average" the color to very different temperatures.
Don't take that the wrong way - Canon AWB is very, very good, especially on the 5DIII and the 1DX. But it is a tool that the user has to understand to get it to work properly.
2) Then you need to understand the Picture Profiles and set those correctly to get **the color that you want**.
The Picture Profiles on Canon are Landscape, Portrait, Neutral, Faithful, etc. I use "Neutral". They will all give you a slightly different look. If you have Lightroom, shoot a RAW image, then flip through all of the Profiles in Lightroom and you will see how the image changes. You are in control of picking that look for the effect that you want.
For the most control, you should be shooting RAW all of the time (I shoot RAW + JPEG and have shot nothing else for 10+ years, even family snaps.) Then you can apply the correct Profile in Lightroom or whatever tool (and set you White Balance there.) BUT, you still should set the custom White Balance in camera.
Even with the Picture Profile set to "Neutral", I still dial in -4 on "Saturation", - 2 "Contrast" and turn off sharpening for the JPEGs when shooting RAW + JPEG, and to set up the look on the rear LCD and the Histogram on the camera.
3) Finally, if you want to control color for best representation for food shots, etc., you should profile your camera using a tool like Color Checker Passport to get to a true neutral color balance. Just like you **MUST** profile your monitor and printer for accurate results, the camera also should be profiled to control device-to-device variation.
If that is way more than you need right now that is fine. The 3 types of color control go from "global" and basic to more refined.
I just wanted to head off any perception people reading this thread might have that the 5DIII is not good on color, AWB, etc. Basically these are all tools that require a certain understanding to get the results that you want. Nothing wrong with that, and it is a learning curve for us all. But the tool is state of the art!