Geometric triangles seem to be quite hot these days.
Source Pixeden
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
Prismatic Polka Dots Mark II No Background
Source GDJ
Light square grid pattern, great for a “DIY projects” sort of website, maybe?
Source Rafael Almeida
A rusty grunge background for websites. Feel free to use it in your site's theme.
Source V. Hartikainen
A seamless chequerboard pattern formed from a tile that can be had in Inkscape by selecting the rectangle and using shift+alt+i. Alternative colour scheme.
Source Firkin
To get the tile this is based on, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Light gray version of the Binding pattern that looks a bit like fabric.
Source Newbury
Background formed from the original with an emboss effect
Source GDJ
Alternative colour scheme. Not a pattern for fabrics, but one produced from a jpg of a stack of fabric items that was posted on Pixabay. The tile that this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
A slightly more textured pattern, medium gray. A bit like a potato sack?
Source Bilal Ketab
This one could be the shirt of a golf player. Angled lines in different thicknesses.
Source Olivier Pineda
From a drawing in 'In an Enchanted Island', William Mallock, 1892.
Source Firkin
This is a seamless pattern of regular hexagon which has a honeycomb structure.
Source Yamachem
Farmer could be some sort of fabric pattern, with a hint of green.
Source Fabian Schultz
Looks like a technical drawing board: small squares forming a nice grid.
Source We Are Pixel8
Heavy depth and shadows here, but might work well on some mobile apps.
Source Damian Rivas
A playful triangle pattern with different shades of gray.
Source Dimitrie Hoekstra