This article shows the first parametric study on turbulent multi-jet impingement cooling flows using large-eddy simulations (LES). We focus on assessing the influence of the inter-jet distance and the cross-flow conditions on the heat transfer at the impingement wall. The LES setup is thoroughly validated with both experimental and direct numerical simulation data, showing an excellent agreement. The inter-jet distance effect on the heat transfer is studied comparing three different distances, where the full Nusselt number profile decreases in amplitude when the jet distance is increased. To evaluate the cross-flow effects, we prescribe both laminar and turbulent inflow conditions at different cross-flow magnitudes ranging between 20% and 40% of the impinging jet speed. Large cross-flow intensities cause a jet deflection which reduces the maxima in the Nusselt number distribution, and it increases the heat transfer in the areas of the wall less affected by the jet impingement. Adding realistic turbulent fluctuations to the inflow enhances the cross-flow effects on the heat transfer at the impingement wall.