<p>We consider compactifications of higher-dimensional supergravities on Riemann-flat manifolds of dimension <i>d</i> (<i>d</i> ≤ 7) that fully break supersymmetry at the classical level on a resulting <i>D</i>-dimensional Minkowski space. We systematically discuss consistency conditions, the Kaluza-Klein (KK) spectrum and harmonics, and the resulting one-loop effective potential <i>V</i>₁, focusing for illustration on maximal supergravity and <i>d</i> = 3, in particular on the 𝕋³/ℤ₃ and on the Hantzsche-Wendt manifolds. We show how the KK spectrum can be re-organized in multiplets of the broken <i>D</i>-dimensional supersymmetry, derive new universal supertrace mass relations valid at each redefined KK level and obtain an analytic finite expression for <i>V</i>₁ after resumming the contributions of all KK levels. In all examples <i>V</i>₁ is negative definite and scales with inverse powers of some internal radii. We extensively comment, when applicable, on the relation with the Scherk-Schwarz mechanism and with supersymmetry-breaking string compactifications on freely acting symmetric orbifolds. We also finally clarify the assumptions and constraints under which Scherk-Schwarz reductions correspond to twisted tori compactifications.</p>