First of all, you don't even have to knit three or four colors per row. Instead you can just knit with two colors and later go back and duplicate stitch the third color. Or you can just knit with two colors, slipping all the stitches that are to be knit with the third color. Then you go back and knit the exact same row again, slipping the stitches you just knit and knitting the third and fourth color stitches you slipped previously.
Still, if you want to knit 3 colors per row there are many, many ways to do this. All are a variety of the methods I've shown last week. You can put all three yarns in your left hand, 3 yarns in your right hand, or 2 yarns in one hand and 1 yarn in the other hand. In knitting with 3 colors a row you don't even have to worry about finding a method that doesn't tangle because they all do as far as I can tell. I generally stop a few times per row to detangle the yarns when knitting 3 color rows.
Here's how I hold the yarns for a 3 color row. The background color is again in my right hand and the other two colors are both stranded over my left index finger.
I knit the yarn in my right hand English/American style and grab the specific yarn I need from my left hand finger and knit continental. This method is slow-going though.
I plan to try the method shown above the next time I have a three color row. I'll put all three colors in my strickfingerhut/knitting thimble and simply use the right hand needle to grab the color I need. I haven't experimented with this method much but I do think it will reduce tangling and the tangles that do occur will be easier to deal with.