Here's the finished coat made from the green and blue random warp I finished weaving this fall. The fabric went of the the tailor in October and came back a few weeks later, but was completely the wrong size. After hemming and hawing and facing the sheer terror of possible total project disaster, I ended up taking it to someone else for alterations. I am pleased to report total satisfaction with how it turned out. I weave on a loom with a sectional beam. Sectional warping was the greatest thing to ever happen to my weaving, and I think it's the only way to do a totally random warp - please post in the comments if I'm wrong on that point.
The down-side to sectional warping is that it takes a big up-front investment in equipment. You have to have a warping box and a spool rack, and it's also really good to have a yardage counter. If you buy these things new, they can set you back close to $1000. Check your local weaving guild newsletter for classifieds of equipment for sale. You can often find great deals, and weavers generally love passing on their stuff to other weavers.
Sectional warping involves winding the beam one section at a time. It lets you control what colors you use in each area of your warp, and can be a fantastic stash buster. I use a mix of weaving yarns, knitting yarns, and my own handspun in my warps. I mix different weights and fiber content, and the only problem I've ever run into is with slubby linen - it did not want to make nice with all the wool I was using. Lesson learned - a little bit of elasticity goes a long way.
To create a warp, I first pull out all the yarns I want to use and wind them onto spools. I put them on the spool rack organized by color to make a palette. I thread the colors I want to use in each section into the warping box, and wind five or six sections at random on the beam. Then I change colors, and repeat this process until the whole beam is full. That's it - a super simple technique that lets you play with color and uses up odd balls of yarn.
One last tip for all the fiberistas who want to try weaving but don't know which kind of loom you want, or aren't ready to invest: check if your local guild has a classroom facility. You may be able to rent time on their looms to figure out your favorite.
The Handweavers Guild of America has a list of local guilds by country here.