This seamless background image should look nice on websites. It has a dark blue gray texture with vertical stripes, it tiles seamlessly and, like all of the background images here, it's free. So, if you like it, take it!
Source V. Hartikainen
Abstract Geometric Monochrome Pattern Prismatic No Background
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
Nothing like a clean set of bed sheets, huh?
Source Badhon Ebrahim
A free background tile with a pattern of pink bump dots. This background tile is sweet! Moreover, it's designed for use as website backgrounds.
Source V. Hartikainen
A subtle shadowed checkered pattern. Increase the lightness for even more subtle sexiness.
Source Josh Green
This tiled background comes in red and consists of tiles that look like gemstones. It is more for blogs or social profiles, I think.
Source V. Hartikainen
To get the tile this is based on select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
From a drawing of the coat of arms of the Ottoman Empire on Wikimedia.
Source Firkin
Simple wide squares with a small indent. Fits all.
Source Petr Šulc.
Medium gray pattern with small strokes to give a weave effect.
Source Catherine
The tile this fill pattern is based on can be had by using shift+alt+i on the rectangle.
Source Firkin
This is a remix of "blue wave-seigaiha".I hope this subtle color version of Seigaiha would be suitable for background .
Source Yamachem
Seamless tile drawn in Paint.net and vectorised in Vector Magic.
Source Firkin
A nice looking light gray background pattern with diagonal stripes.
Source V. Hartikainen
From a drawing in 'Cowdray: the history of a great English House', Julia Roundell, 1884.
Source Firkin
One can never have too few rice paper patterns, so here is one more.
Source Atle Mo
From a drawing in 'Danmarks Riges Historie af J. Steenstrup, Kr. Erslev, A. Heise, V. Mollerup, J. A. Fridericia, E. Holm, A. D. Jørgensen', 1897.
Source Firkin
Remixed from a drawing in 'Очерки Русской Исторіи въ памятникахъ быта', Petr Polevoi, 1879.
Source Firkin
This pattern comes in orange, and it looks as if it is "made of glass".
Source V. Hartikainen