A structural phase transition from cubic F d3m to tetragonal I41/amd symmetry with c/a > 1 is observed at TS = 16 K in spinel GeCo2O4 below the Néel temperature TN = 21 K. Structural and magnetic ordering appear to be decoupled with the structural distortion occurring at 16 K while magnetic order occurs at 21 K as determined by magnetic susceptibility and heat capacity measurements. An elongation of CoO6 octahedra is observed in the tetragonal phase of GeCo2O4. We present the complete crystallographic description of GeCo2O4 in the tetragonal I41/amd space group and discuss the possible origin of this distortion in the context of known structural transitions in magnetic spinels. GeCo2O4 exhibits magnetodielectric coupling below TN. The related spinels GeFe2O4 and GeNi2O4 have also been examined for comparison. Structural transitions were not detected in either compound down to T ≈ 8 K. Magnetometry experiments reveal in GeFe2O4 a second antiferromagnetic transition, with TN1 = 7.9 K and TN2 = 6.2 K, that was previously unknown, and that bear a similarity to the magnetism of GeNi2O4.