Dry cleaning.
OR
Find a friend with a sloping paved driveway.
It must be spotless.
Flip the rug over and hose it down.
Gently massage in a mild solution of Woolite and cool water.
Rinse.
Hang to dry for a few weeks.
I actually did this (with great results) with a fragile and already slightly fraying antique oriental rug.
I used a thick dowel covered with an old towel suspended in a friend's garage where it dripped and air dried for a couple of weeks.

My washer and dryer are large front-loading (energy efficient for multi-generational family under one roof!) and I actually put my thrifted chenille rug in there. Mine turns out great, but it's pastel colored, and your bright colors might be another consideration.

In the nice weather, I hang it over the bannister of my deck to save energy. Shake to fluff.

And yes, it fills the lint screen, but I've had mine for years.

I would think a hand-held rug cleaner (with no agitator) would work well.

I use my Bissel Little Green Deep Reach (http://www.bissell.com/little-green-proheat-deep-reach/) on a ton of different things (pretty much everything but my boyfriend at my dog, at this point) and had a lot of luck. Especially with smaller, hand-made rugs I've found in thrift stores, etc, I've had little to no trouble with colors running or lots of fluff coming off. I don't use the chemical stuff it comes with, just hot water, and it works like a charm. Well worth the investment.

Those look exactly like one I inherited from my mom when she moved out of her house. She'd had it for so long, all the fringe was worn off. I used it in my kitchen for a couple of years, and washed it in my front loader several times. I hung it up to dry. I never had any problems with colors running, or a lot of lint, or shrinkage. I expect my mom washed it regularly, too.

I'd wash them in cold on the gentle cycle. If you're worried about keeping the colours from running, add a cup of salt to your laundry detergent.

My first line of action would be a thorough vaccuming using a hand brush. Sit down and use a brush on the hose and take your time going over it, front and back. Make sure you don't fold dirty parts onto clean parts as you work.

If that doesn't cut it, then I'd try washing them in your bathtub. No agitation, twisting, wringing, etc. fold it accordion style, make a good amount of mild bubbles, and gently sponge the carpet with a clean car washing sponge. Use cold water to prevent dye bleed. You'll want to transfer it to a clean sheet to roll it and move it. Ideally you could dry it laying flat on a porch or something on a nice day.

Tumble wash one at a time in one of the large washers at a laundromat. Cold water. Tumble dry one at a time on warm/cool setting until until dry. No bleach, no liquid fabric softener. Enjoy!

Looks like veslabeachgirl got to it before me - I was also going to suggest the "Take it outside and hose it off" option. If the driveway isn't anywhere near spotless, a tarp might be necessary. Or just hose it off and wash while it's hanging on your fence. It's probably going to have to dry there anyway.

I wouldn't put them in the washer.

I suggest hand washing in cold water with Woolite.