These types of tablecloths are fairly prevalent in British Cafe's, although usually made of plastic.

Barring any relation to Italian culture specifically, the cloth that you are referring to is known as 'Gingham' and was originally two separate fabrics of different colours woven together, as opposed to dyed afterwards.

This book references Gingham fabric as fashionable in the 18th Century, whereas there is fairly comprehensive information placing Ginghams origins in the 1600's with its importation to the UK through the East India Company here.