If you have any tall grasses that wintered over, be sure to let them be. Same with small twigs and vines. Birds will naturally go to them.
Although some folks say don't put out artificial fabric, I have found the birds will use polyfil fiber. (Mr. Squirrel chewed one of our cushions for nest material, and we saw it later in the nests of our bluebird houses!)
You can try shreds of cotton cloth - flannel is popular, but make sure the cloth is worn enough that the birds can take shreds from it. Tear strips about 1/4 inch wide by 1/2 to 1 inch. Oh and before you tear it up launder it really well in NO SOAP and no softener.
Real cotton - Wadding if you can find it in the first aid section of the drugstore, otherwise cotton balls, are very popular, as are bits of yarn.
We also like to leave some bits of horsehair (f you have a friend who owns a horse, ask for some mane cuttings) and oddly, many wrens and finches seem to enjoy bits of cat hair that are on the cats beds that stay on the porch!
Anyway, place the bits out where the birds will safely find them, you can use an inexpensive suet feeder and fill it with different bits. (change them out after a bunch of rain where they don't get a chance to dry)
We also stick some in cracks and crevices on the edge of the deck, a handrail, even in the bark of our hickory tree!
Put them in different areas and watch for activity, put them high and low, and near your feeders so the birds find them.
We are just outside of Chattanooga TN, and I put our stuff out about two weeks ago; already there is activity. Both of our bluebird houses are already claimed!
If you are zone 6 on the east coast, and really want to have fun watching nesting birds, I suggest a wren house and a bluebird house! Put them up facing east and within a few days to weeks you will see someone moving in!
good luck and have fun!

Excelsior (would be plentiful now, as it is often used as a natural filler for Easter baskets), jute, cotton string, raffia. All of these materials are perfect for nesting material, readily available, and easy to prepare.

I've noticed that the robins, cardinals, and wrens are getting territorial, so I'll be putting some material out this week, probably.

Just be careful what you have in your area. Last year a robin put her nest in the holly tree. They used all kinds of things including rope which was fine. However they also used part of the undermat from the sod which had been laid a couple of years ago. This is like monofilament. One of the babies wrapped it around its leg above the point where feathers start. I did not find it until it was too late.

Some used to suggest laundry lint but I have not noticed it in recent recommendations.

I read somewhere that smaller birds collect thistle down and the fluff from milk weed pods. So, I saved what I could when harvesting seeds. I never actually saw who took it (the wind?); but, it all disappered much faster than the dog's undercoat and cotton I provided.

I put out the hair I pull out of the hairbrushes-ours and the dogs'. Soft and safe.

There are many ways to attract birds, but as far as nesting materials go, I put out some cat fur every Spring. The birds love it! My sister has long-haired cats and she saves some of the fur for me. In the Spring (usually around March or April?) I fill a suet cage with the cat fur and hang it from a tree. Last year it was the large Oak tree near the deck and bird feeders. Within a few weeks, the fur is gone! Assorted birds take the fur and even a red squirrel.

Here is a Titmouse getting fur a few years ago -
{{gwi:314214}}

Here's the red squirrel last year -

do birds use nesting material in the winter to stay warm? should I put out my dogs' hair now around Thanksgiving?

You can buy (or make) winter roosting boxes. My SO made me one. It's essentially a nesting box with the opening at the bottom. (to hold in the heat) He also put in 2 alternating rows of perches inside. (so more can fit in.)