I get queries all the time about where to purchase fabrics appropriate to the period. So this blog is about giving a regularly updated listing of the best, based on quality & customer service rather than price alone. They are all also availble for purchase via the internet, which means they are accessible no matter where you live.
Linen
Without a doubt, Fabric-store.com should be your first stop. They offer a discount for repeat custom. They are quick, reliable, and carry linens in a good range of colours and weights, including linen canvas, which is not exactly easy to find. I highly recommend getting a fabric sample first, as the colours on the website are not necessarily exactly true.
Wool
Friends have had luck with B. Black & sons for fine quality wools woven in period styles in a range of colours. For a truly authentic fabric, Time Warp textiles will weave reproduction wool fabrics to order. To be honest, my stash is large enough that I haven't bought wool in some time, so your mileage may vary.
Silk
For silk brocades try Sartor - all about silk. While they supply a number of silks, of particular interest is the brocades section. You need to have researched some of your fabric patterns first, but you will find replica fabrics of period textiles. For liturgical or geometric silks, try Istok. Be aware that some of their silk fabrics are rayons - the price is usually a good indicator. Order a sample and test the fabric first before you purchase.
If you have a particular pattern in mind for a fabric and can't get it at either of the above silk merchants, then getting a fabric custom printed might be the right alternative. Try Spoonflower. They even offer it in a range of silks. You could also try M. Perkins & Son, who offer a range of partial silks or silk substitutes in beautiful patterns.
Unusual Fabrics
For both hemp and buckram, which are appropriate, but oh-so-difficult-to-find, try Wm. Booth, Draper, a reenactment specialty supply store. Again, I have not used them, but I plan to in the future.
I hope this helps - any other suggestions are more than welcome.