There's a trick to this type of needlepoint stitching if you are willing to have a few needles going at once. Here's how:

Needlepoint Stitching With Multiple Colors

Thread a needle with each of the repeating colors e.g. if there are three repeating colors then thread three needles with one of the colors on each.

Then start the second color (secure the starting "tail" under some other stitches), and stitch along until you either reach the first color (the white), or if the pattern has more than two repeating colors, until you come to the end of this second color. Then take the needle and insert it into the beginning of where you will use this second color again.

Repeat this process along the row, using all the colors. You will see in the photo here we have two colors and the orange is approaching the white. When we reach the end of the orange we will place the needle up into the hole where the orange will start again and then start stitching with the white.

This needlepoint stitching technique means that the threads are dragged across the back of the canvas with each change of color, but they are then covered up by the stitches that are created afterwards. So the back of the canvas actually looks nice and neat. It also means you are not having to tie off all the time as you keep two or more needles on the go at any one time. So, it's a faster way of needlepoint stitching a repeating pattern.