Determine the age and origin of the furniture, which can provide an indication of what wood it most likely is. The older it is, the more likely the wood is oak, not walnut.
Look at the color of the furniture, and remember that it might be stained. Old antique oak is a dark brown, while old walnut may have a more golden tinge. Walnut is sometimes stained a reddish color to resemble mahogany.
Examine the grain, which is the key difference between the two. Walnut has a fine grain, with the lines in the wood that come from the annual growth rings of the tree being very close together. Oak has a wide, open grain, with the lines sometimes being 1/2 inch or more apart.