The Jahn-Teller effect is a fascinating and ubiquitous phenomenon in modern quantum physics and chemistry. We propose a class of oxides with melilite structure Ba$_2$TGe$_2$O$_7$ (T=V,Ni) where Jahn-Teller distortions are the main responsibles for the appearance of electric polarization. At the hearth of the proposed mechanism lies the lack of inversion symmetry displayed by tetrahedrally coordinated transition-metal ions, which allows for the condensation of polar Jahn-Teller distortions, at odds with octahedral coordination typical of conventional ferroelectric oxides with perovskite structure. Since the noncentrosymmetric local environment of transition-metal ions also activates the proposed spin-dependent hybridization mechanism for magnetically-induced electric polarization, proper multiferroic phases with intrinsic magnetoelectric interaction could be realized in this class of low-symmetry materials.