edit//PLEASE NOTE! I brainfarted while doing these, the last month at the right end of the table is October, not December! Of course, don't let this keep you from wearing hitoe further into the autumn if it's too hot for you ^_^;
So a certain wonderful someone sent me the Summer 2009 edition of Utsukushii Kimono, and I thought I'd share the amazing super-detailed summer weave usage chart found inside. Prepare yourself for a million usumono terms!

Here be a ~10Kb version for modem users.
Notes for most exotic terms:
透ける織物, literally "see-through weaves"
献上 kenjô
塩瀬 Shioze (a kind of taffeta?)
I found a lovely picture that shows some fabric types! Click to see.
List of terms used and what they are/mean/I think they are, in alpabetical order:
Asa - hemp. Also used for many other non-silk fibres (linen, flaxen) other than wool.
Hitoe some/ori - unlined, patterns dyed/ patterns woven
Namatsumugi - means "raw tsumugi". A picture I found. It looks like rough tsumugi.
Ra - a kind of mesh. As far as I know, it looks like "silk kôbai" in the picture linked to above, but patterns are sewn on like in embroidery. A picture is posted below.
Ro - ro, cloth with horizontal lines of tiny holes (or vertical, if it's "tatero")
Ro chijimi - ro crepe: crepe cloth with horizontal lines of tiny holes
Ro shioze - a type of ro fabric. Shioze seems to be a fine, plain, unpatterned type of silk cloth (a picture of a man's han'eri I found)...
Ro tsuzure - it seems to be a kind of non-fancy weave (click for picture or "tsuzure"). Again, with tiny holes.
Sha - like ro (with tiny holes in the fabric) but the pattern is like a grid.
Sha awase - lined with sha fabric (see keyword)
Sha kenjô - it seems to be a kind of weaving used in hakataorimono, where the pattern is formed by weft that is left unwoven for certain lengths. With tiny holes.
Sha tsumugi - tsumugi, with leeeeetle holes
Sukeru orimono - literally "see-through weave", I think it covers every other type of tee-through fabric there are

Pictures of: ro, ro tsuzure, sha, sha tsumugi, ra, sha kenjô, asa, ro shioze.