I agree with MadQuilter -- Cindy Needham does this kind of stuff all the time. She backs the linen with a fusible 100% cotton woven interfacing called Bosal #300. I have seen it used by members of my local quilt guild with linen & it looks absolutely stunning. I highly recommend her iQuilt classes. She works with linen wholecloth fabrics (like tablecloths), but I've seen people who do mix & match piecing. You may want to get a small amount of scrap linen fabric from the store to practice with the Bosal. Some people end up pressing those seams open -- I'd just experiment both ways.

The other option you have would be to applique the linen on top of a piece of cotton. In that case, I would recommend a fine cotton or silk thread for the applique work in a matching color to the linen (Aurifil 50/2 is great because it comes in so many colors -- or else YLI silk thread is a good option as it is very fine). The advantage of cotton thread is that even after mercerizing, it tends to be rather dull -- like linen. The advantage of silk is that you can get a 100wt thread that, if a near match in tone to the linen fabric, will be nearly invisible.