Cubic spinel chromite ZnCr₂O₄ is a geometrically-frustrated antiferromagnet with the Neel temperature T_N = 12.5 K, and the Weiss temperature θ_W = -390 K. The magnetism of this compound is fully dominated by the orbital-inactive Cr³⁺ (3d³) with spin S = 3/2, which forms a sublattice of corner-sharing tetrahedra (pyrochlore lattice). We study effects of element substitutions on the geometrically-frustrated magnetism of ZnCr₂O₄ by investigating structural and magnetic properties of polycrystalline Zn(Cr_1-xFe_x)₂O₄, Zn(Cr_1-xV_x)₂O₄, and Zn(Cr_1-xMn_x)₂O₄. For these mixed crystals, the substituted Fe³⁺ (3d⁵) with S = 5/2 is an orbital-inactive ion, but V³⁺ (3d²) with S = 1 and Mn³⁺ (3d⁴) with S = 2 are orbital-active ions. The experiments reveal that the antiferromagnetic order of ZnCr₂O₄ is sensitively suppressed by the element substitutions, and instead, the spin-glass-like behavior appears at low temperatures below ~ 10 K. This result implies the competition of geometrical and bond frustrations in Zn(Cr_1-xFe_x)₂O₄, Zn(Cr_1-xV_x)₂O₄, and Zn(Cr_1-xMn_x)₂O₄.