Okay, I think I do have an answer now. I'm borrowing arguments from the proof of Lemma 2.27 in the book "Fourier Analysis in Convex Geometry" by A. Koldobsky (apparently not available online at all). That lemma states that the Fourier transform of the function (on R) exp(-|x|^p) is positive everywhere for p in (0,2].
The central tool is a theorem of Berstein, which in particular implies that if s is in (0,1] then exp(-z^s) is the Laplace transform of some finite positive measure \mu on [0,\infty); that is, exp(-z^s) = \int exp(-uz) d\mu(u). Applying this with s=1/p and z=||x||_p^p yields exp(-||x||_p) = \int exp(-u ||x||_p^p) d\mu(u).
Now calculate the Fourier transform on R^n of this. Using Fubini you get an integral wrt \mu of a product of Fourier transforms of exp(-|x|^p), and you can now apply the one-dimensional lemma. (The one-dimensional lemma is proved by using the same theorem of Berstein to reduce to the case p=2.)