We show that every closed spin manifold of dimension n ≡ 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheres Sⁿ, n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sum M₁ # M₂ with certain metrics is close to the union of the spectra of M₁ and of M₂.