How about silk knit? It's got the texture thing going for it, and silk knit ties tend to be more casual than most. Since linen summer suits also tend to be relatively informal (by suit standards), the pairing of line suit with silk knit tie would seem a good one.
A silk grenadine tie might work, too. (Even though such a tie is not, generally speaking, considered to be a particularly casual tie. But its texture would work with a linen suit.)
Anyway, I don't understand why you'd steer away from silk, for textural reasons. The texture of a "normal" (non-knit, non-grenadine) silk tie can definitely differ from that of a linen suit. Thus, you have contrast between the two textures. And that's presumably what you want. Nor does the contrast have to be night-and-day. A more subtle difference is often perfectly adequate.
I agree that cashmere has its own texture, but I just don't see it as a great summer tie. There are alternatives to cashmere (some of which I've mentioned), so I'd recommend steering away from cashmere.
Grenadines can run up and down the formality spectrum pretty easily, though. They're sort of unique that way among ties. For a more casual or rusticated look, you could go with an unlined and untipped grenadine in a looser weave, like garza grossa.