These first two pictures are from Il Cuoco Segreto di Papa Pio V (The Private Chef of Pope Pius V), by Bartolomeo Scappi, Venice, 1570. As you can see, they are pierced metal.
This detail is also from Il Cuoco Segreto di Papa Pio V (The Private Chef of Pope Pius V). I've left the man in there so you could see the relative size of the strainer. This large cone is probably made of cloth and is used for draining curds.
The Sievemaker is from Eygentliche Beschreibung Aller Stande auff Erden by Jost Amman and Hans Sachs, Frankfurt, 1568. His sieve frames appear to be made of wood, like Shaker band-boxes. The tall things in the background, according to the caption, are thin strips of wood, which he weaves together to make the strainer part. These sieves are suitable for dry things such as flour and spices.