Since you're working with 3 colors, you will have to cut eventually.
If you want to carry up the side, you'll have to work two rows with one color (Red, for example). Once the second row of red is done, just let the red hang loose. Add another color (green, for example). Knit two rows with green. To "carry" the red, just pick the red again and start knitting. I'm not sure what the maximum rows for carrying is, but I think it's four rows.
I once knit a pink and mauve scarf out of scrap yarn; it's one of my favorite scarves The only cutting I did was at the end of the scarf because I worked 2 rows of pink, 2 rows of mauve ... then I just picked up the pink again ... mauve ... pink ...
I hope that helped a little... :?