Yeah, woven seem good, or a linen.

I have quite a few sets (not round) for my kitchen table. You might want to try these "wedge" placemats.

wedge placemat

We have a 44" round table, and use these:

https://www.houzz.com/products/oval-rattan-placemat-set-of-2-honey-brown-prvw-vr~4714570

Do you have a pic of the kitchen that shows your colors in there such as window treatments, wall color, so we can take them into consideration?

I use oval on my round breakfast room table.

I have some of the wedge shape that torreykm linked to. I also have some round IKEA placemats that seem to work. No awkward corners.

I have oval Chilewich placemats on my round/oval table. Love them, so easy to clean and they're (IMO) more stylish than some other vinyl placemats. Crate and Barrel has them, I think, and maybe Amazon too.