washing Girard pillows  

Does the fabric have a zipper?
If it does and you can get the fabric off, great.
I'd recommend Woolite in cool water. It's a gentle detergent and it rinses easily.
If you use a regular detergent, it'll take forever for you to get the soap out.
So...soak it in Woolite, carefully roll it up and remove alot of the soap, then rinse it in cool, plain water, and then roll each one in a proper terry towel rather tightly.
Then lay it flat, or suspend it vertically. It shouldn't shrink...

Thanks! And, yes the covers...
Thanks! And, yes the covers come off. It is more like a pocket with no zipper where you just pull the pillow out.

You could send them to the
Dry Cleaners too, but my personal opinion is that I don't care for the smell of the chemicals they use and fabrics can get 'crispy'.
I'd try Woolite first, although I cannot guarantee that the fabrics might shrink a bit....

The tag says not the dry...
The tag says not the dry clean - which I though was weird.

Maharam recommends
dry cleaning, since the fabric is 'upholstery fabric' (their term).
Other than the possible shrinking issue, I fail to understand why gentle washing with Woolite wouldn't work nicely,

I probably should have mentio...
I probably should have mentioned mine are from Herman Miller for the Home ansd Maximo. I think the Maharam fabrics are all woven patterns, whihc should be more resilient than these screenprinted illustrations.

Herman Miller pillows
should all be made of Maharam fabric since Herman Miller's Girard fabric is being made by Maharam....I think.

Shrinking shouldn't be an iss...
Shrinking shouldn't be an issue if you use lukewarm water and dry it away from heat. Most shrinking occurs as the fabric dries, especially when it's drying in a hot clothes dryer.
This particular fabric's inks are probably soluble to some degree in drycleaning fluid, which is why it says to avoid drycleaning.
Washing in a washing machine could cause creasing during the spin cycle and that might leave permanent lines in the screen printed areas.
If the white cotton fabric is really grimy and doesn't come clean with a soak in mild detergent, you can try Oxy-Clean. I've used it with great results on antique linens and clothing so it should be safe for your fabrics. It breaks down body oils and leaves the fabric very white. You do have to soak the fabric for up to six hours, though. And gently push it down in the soak after 10-15 minutes and then again a bit later because oxygen is released and will float the fabric up to the surface. You want it to be submerged.
A little white vinegar in the rinse water will help to get all the detergent out no matter what type you use.