This is the front side of my cable. As you can see, I'm doing a simple rope cable, with one half of the cable in grey and the other half in navy, on a burgundy background. I've got 4 knits and 2 purls in the burgundy, then 4 knits in the first half of the cable in either navy or grey (depending on which row you're looking at), then 4 knits in the second half of the cable in the other color, then 2 purls and 5 knits (because this is the band for a hat and I'm going to lose one stitch behind my pickup row when I start the top) in the burgundy again. There are 8 total stitches in the cable, and I am crossing the cables every 10 rows.
Here is the back of my cable. You can see quite clearly here that I'm using intarsia techniques to do the colorwork. Each of the burgundy strips, and each half of the cable, has its own strand of yarn, as you can see at the right edge of this picture.
Here's how this works: On the non-crossing rows, I work the first burgundy section, then drop that yarn, and pick up the yarn for the first cable section so that it's coming from below and to the right of the burgundy yarn. I work those 4 stitches, and then drop that, pick up the yarn for the other cable half in the same manner and work those 4, and then I drop that, pick up the second strand of burgundy yarn in the same manner, and work the second burgundy section. As I pick up each yarn I am careful to pull the last stitch from the prior row snug, making sure there is no looseness as I work the first stitch in this row in this color; I don't worry about the snugness of the strand from the section to the right that's being caught, because I will have the opportunity to take up any slack there when I work that color on the next row. I am working directly from balls of yarn, not from bobbins (which I dislike); I keep them straight by turning my work CCW when I go from RS to WS and CW when I go from WS to RS -- if you do this, and you pick up your strands properly, you will tangle them up when you work the RS row, but you'll untangle them perfectly when you work your WS row and all will be well.
So far that's just garden-variety intarsia; the only tricky bit is the crossing row. In these photos, I'm just about to do the next crossing row, where I'll be crossing the grey half in front of the navy half. Here's what I'll be doing. I'll work the 6 burgundy stitches, and then drop that yarn. This time, I won't twist the grey yarn around it; instead, because this is going to be a front cross and I don't want the grey yarn tangled around the navy stitches, I'll pick up the grey ball of yarn, bring it over the needle and set it in front of the work, to get it out of the way. I do my cable crosses without a cable needle, so the next thing I'll do is go into the back of the 4 navy stitches with my right needle and capture those, and then pull the 4 grey stitches and 4 navy stitches off the left needle. I'll re-capture the 4 grey stitches with the left needle, and then I'll pause a second, while I've got the middle of the cable opened up, and I'll move the grey yarn back over the needle, making sure it's not wrapped around either needle tip, and set it down to the left of the navy yarn, and then I'll slip the 4 navy stitches back to the left needle. Now, I'll pick up the navy yarn so it catches the burgundy strand, work those 4 stitches in what are now the first half of the cable, then pick up the grey yarn so it catches the navy and work what's now the second half of the cable, and then pick up the second burgundy strand so it catches the grey and work the final 7 stitches.
Here's a couple of variations:
1. What if I did use a cable needle? In that case, I would still move the grey yarn forward, and then I would slip the 4 grey stitches to the cable needle and hold it in front. Then I'd catch the burgundy yarn with the navy, and knit the 4 navy stitches, and then I'd put the grey stitches back on the left needle, and move the grey yarn back, and then catch the navy yarn with it and work the grey stitches. Everything else would be the same.
2. What if I was twisting the other way, so that it was the navy half coming forward? In that case, before moving stitches, it would be the navy yarn I'd bring to the front, and then I'd come into the front of the navy stitches with my right needle, slide all the cable stitches off my left needle, and recapture the grey half of the cable, and then go ahead and put the navy stitches back on the left needle also. Since the front-half yarn isn't in the middle of the cable, I don't need to move it back until I've done all my rearranging, and then I'd simply proceed with knitting across all the stitches in their respective colors as usual.
3. What if both halves of my cable are the same color? This question is more complex than it looks. At first blush, it seems that you'd simply have one strand of the cable color, and two strands of the background color, but look at where your yarn ends up. If you have one strand of the cable color, it's attached to the rightmost stitch of the cable. When you do your rearranging, it ends up in the middle of the cable, and you have to strand it back to the new rightmost stitch in order to be able to wrap it around the background strand and to start knitting that stitch. This is awkward at best, and likely to lead to weird pulling, looseness at the join between the background and the cable, or both. Instead, I'm going to suggest that it's best to have one strand for each segment of your cable (so 2 for a simple rope cable like this, 3 for a 3-strand braided cable, and so on), so that each segment can take its own strand of yarn with it when it moves. This does give you additional intarsia joins in the center of your cable, but they should be invisible if you're doing your intarsia right, and I think the results are worth the small additional hassle.
Do you have to use intarsia to do multi-colored cables? Nope. With a suitable project, such as small cables with the background color interspersed frequently, you can use stranded knitting -- Eunny Jang's Chuck's Cabled Socks, for instance, are stranded, and are a perfect example of where that's suitable. However, the usual caveats of stranded knitting apply -- short color segments (in both the cables and the background) to avoid long floats are generally required, and you get the usual characteristics of a stranded fabric, which is thicker and less stretchy and experiences flattening of knit-purl textures, whereas intarsia fabric has the same feel and behavior as single-color fabric. I think intarsia is, in most cases, going to be your best choice.
Edit: As faecat and I discuss down in the comments, in some circumstances a blended approach makes sense. When you have small, separated cables, as ones sees in some Celtic designs, and you're doing separate colors for each strand, as she has done in her gorgeous multi-color Rogue sweaters, then stranding the background color but using separate strands for each of the cable colors works beautifully. With this, because you're not drawing strands behind the purled sections, you don't get the flattening that a pure stranded approach would give you, and because most of the fabric is not stretch-limited by a straight strand, you retain a fair portion of the stretch you'd have in a single-color fabric.