With[{z = 3 Pi Exp[I th]},NIntegrate[z^2/(Exp[z] - 1)^2, {th, 0, 2 Pi}]]

N[2 [Pi] I
   (Residue[z^2/(Exp[z] - 1)^2, {z, 2 Pi I}] +
    Residue[z^2/(Exp[z] - 1)^2, {z, -2 Pi I}])]

With[{z=3 π Exp[I θ]},
    NIntegrate[z^2/(Exp[z]-1)^2 D[z, θ], {θ, 0, 2 π}]
]

NIntegrate[z^2/(Exp[z]-1)^2, {z, 1, 3 π I, -1, -3 π I, 1}]