Why walnut is a different stump from the rest
Black walnut is on every fencerow and field edge in this county and it is a genuinely valuable tree, which is why so many stumps around here are walnut rather than the whole tree. Somebody sold the log. What is left is dense wood that grinds slowly and a root system that has spent decades putting a compound called juglone into the soil around it.
Juglone is the tree's own herbicide. It suppresses competition, which is why the ground under an old walnut is usually bare or thin. The highest concentrations sit in the roots, the hulls and the bark, which means the material a grinder produces is more loaded with it than the trunk wood alone would be. That is the single fact that makes walnut spoil different from maple spoil, and it is the reason this question comes up every summer.
What it does and does not kill
The plants that genuinely fail near walnut are worth knowing by name, because the list is shorter than the internet suggests. Tomatoes are the classic and they wilt dramatically. Peppers, aubergine and potatoes react as well, and so do apples, blueberries, azaleas, rhododendron, lilac, peonies and white birch. If your vegetable patch is going where a walnut stood, this is not a superstition.
What does fine is most of what people actually plant in a Trousdale yard: fescue and the usual lawn grasses, maple, oak, hickory, redbud, hackberry, cedar, most viburnum, daylilies, hostas, ferns, black-eyed Susan, beans, carrots, onions, beets and corn. The sensitivity is species-specific rather than general. A lawn going back over a ground walnut stump is not a problem. A raised bed of tomatoes in the same spot next April very much is.
How long the grindings stay a problem
Juglone breaks down with oxygen, moisture and microbial activity, and the pace depends entirely on how the material is stored. Spread thin in a sunny place and turned occasionally, walnut grindings are broadly safe to use within two to three months. Piled two feet deep in a shaded corner and left, they can stay potent for the best part of a year, because the middle of that pile is anaerobic and nothing is breaking down.
So the practical advice is about handling, not avoidance. If you want to keep walnut spoil, spread it no more than three or four inches deep somewhere it gets sun, and leave it a season before it goes near a bed. If you want to use it straight away, put it on a path, round the base of established trees that are not on the sensitive list, or on a fenceline where nothing is being grown. Do not put fresh walnut grindings on a vegetable garden and do not compost them into potting mix.
Planting a new tree where a walnut stood
Two separate problems, and people usually only think about one. Juglone is the first. The second is the nitrogen sink: a hole packed with any wood grindings has a carbon-heavy fill and the microbes breaking it down will take nitrogen out of the surrounding soil for a couple of years, which starves a young tree at exactly the wrong moment.
The fix for both is the same. Take the spoil out of the hole rather than backfilling with it, bring in topsoil, and set the new tree three to six feet off the old centre so its roots run into ground that was never worked. If you specifically want tomatoes or blueberries in that area, give it a full season, work compost through, and consider a raised bed with imported soil rather than fighting the ground chemistry. On a lawn, none of this applies: sow fescue over topsoil and it will not care what the tree was.
The short version
Walnut spoil is not toxic waste, it is just selective. Grass, oak, maple and most ornamentals are unbothered; tomatoes, blueberries, azaleas and apples are not. Spread walnut grindings thin in the sun for two to three months before using them anywhere near a bed, and if a new tree is going in, haul the spoil out and plant a few feet off the old centre in fresh topsoil.