How to knit a rag rug:

Cut strips of fabric. You can cut or rip strips of fabric across the grain, but then you’ll have to either stitch them together or work them in like yarn ends, which is just a pain (and in the latter case, very bulky!) If you cut your chunk of fabric like this (lines are cuts):

you’ll end up with one big long strip that looks kind of like this:
(not remotely to scale).

The lumpy bits are the cut-ends where the strip changes direction, or goes around a corner, however you want to look at it. They will make little texture flaps in your finished mat. If this is going to bother you, I’m sorry, you’ll just have to stitch flat strips together. Ditto if you want to get all fancy and work in different colors/fabrics. This can be done with a quick zigzag on the machine, or hand tacked, whichever you find faster and more convenient. Just lap the ends over each other by half an inch or so, and tack them together.

Wind your long strip of rag into a ball, for your own peace of mind (if you are using woven fabric it will also reduce the horrible tangles that come from fraying).

And then: Just start to knit! Garter stitch works fine, and as long as you are content with something “approximately so big”, you don’t even have to swatch. If you’re picky about the finished size, of course, you’ll have to swatch same as anything else, but your swatch will have to be pretty big – four inches isn’t really going to tell you much, as it’ll wind up being only about six or eight stitches wide. Use big needles – 10 mm and up. (What is that in American, size 16? Something big, anyway.) I think I used broomsticks for the bathmat, but I honestly forget. It works with wovens, and they don’t really seem to fray once they’re knitted together – I suppose they would deteriorate eventually, but I put my mat in the washer and dryer all the time with no deleterious effect. It is 100% cotton weave, ripped on grain, and it frayed like the very devil while I was making it. You can do this with knit fabrics too, though that may seem redundant, and if you try it with something like panne velour on 10mm, it is soooo soft and yet so thick and sturdy! Solids will be solid, stripes will give you variegated effects, prints will give you more visible effect the larger they are, but since most prints are darker on the “right” side, be aware: some of the “wrong” side of your original fabric is definitely going to be showing on the “right” side of your knitting, the fabric won’t lie flat and you don’t actually want it to. The exception to this is if you use a knit – it will usually roll under when cut, so you will have mostly the right side showing in your work.

This is my bathmat, knit from 1” wide strips of striped cotton:

The whole thing cutting and all took two, maybe three hours, and having made braided rag rugs before, I can join with the anonymous 19th c. housewife in saying that this is a LOT faster!

A few other notes:
If you are using new fabric, pre-wash it the same as you would for sewing it (you would, right? I thought so.) If you’re cutting up old garments or using scraps, cut to get the longest strips you possibly can, but obviously you’ll have some piecing to deal with. The thinner the strips and the lighter the fabrics the more likely you’ll be able to just knit in the ends – heavier and crisper fabrics, you will notice a bulge where you’ve got double thickness, which is why I prefer to lap and stitch. Cutting fabric on bias hasn’t given me a huge problem, but if you were making something that wasn’t going to just lie on the floor, a coat or a wall hanging eg., it might pull the shape out of true. This problem could probably be solved by knitting the bias fabric together with yarn or string or something to stabilize it.

For a plain square mat, garter stitch gives a good thickness, and a bit of springiness that’s nice to step on. Once you get the hang of the size though, knitting with rags is the same as knitting with yarn, texture stitches, increases and decreases, all the same. So of course, you could make a round rug, or a patterned one, or play with texture like basket weave, or a big cabled border. The details of fancy stitches will not show up particularly well, especially if your fabric is print or stripe – you will probably have better luck with large texture areas. But ribbing or seed stitching, although they may not show up visually as clearly as they would in yarn, will still have the same elastic or stiffening effects as they would in another medium, so they could still have uses!