  We show that, given a set $E\subset \mathbb R^{n+1}$ with finite
$n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E}
f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in
$L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a
compact set $E\subset\mathbb R^{n+1}$ with $H^n(E)<\infty$ is removable for
Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable,
thus proving the analog of Vitushkin's conjecture in higher dimensions.
