Serious Eats - seriouseats.com

@Sheri, you say you don't cook with wine, but it's not clear whether it's wine specifically, or if you don't cook with any alcohol whatsoever... it might be easier to come up with suggestions if we understood the reasons. For instance, if it's because you don't drink wine and therefore the wine will spoil before you can use it up, then sherry or vermouth my be an option. If it's because you just don't like wine, try cognac or brandy.

On the other hand, if it's about avoiding alcohol all together, and if the wine is used to deglaze the pan, or in large quantity for cooking liquid (as in risotto), then I'd go with stock. Juices and vinegars will be way too sweet or tart for those purposes, and will overrwhelm the dish. For white wine, use chicken stock. For red, use a beef or mushroom stock. You can replicate a bit of the acidic quality that would have been gained from the wine by adding a very small amount of mild vinegar (white wine, champagne, rice) or a squeeze of lemon juice. You'll want to use these very sparingly in proportion to the chicken stock.

Also, as mentioned previously, it's not necessary to use any liquid at all with sauteed mushrooms specifically.

Apple juice works some times.

My mom always just sauteed hers with lots of garlic, a splash or two or Worcestershire sauce and some freshly ground black pepper.