Pulsed-field magnetization experiments (fields $B$ of up to 85 T and temperatures $T$ down to 0.4 K) are reported on nine organic Cu-based two-dimensional (2D) Heisenberg magnets. All compounds show a low-$T$ magnetization that is concave as a function of $B$, with a sharp ``elbow transition to a constant value at a field $B_{rm c}$. Monte-Carlo simulations including a finite interlayer exchange energy $J_{perp}$ quantitatively reproduce the data; the concavity indicates the effective dimensionality and $B_{rm c}$ is an accurate measure of the in-plane exchange energy $J$. Using these values and Neel temperatures measured by muon-spin rotation, it is also possible to obtain a quantitative estimate of $|J_{perp}/J|$. In the light of these results, it is suggested that in magnets of the form [Cu(HF$_2$)(pyz)$_2$]X, where X is an anion, the sizes of $J$ and $J_{perp}$ are controlled by the tilting of the pyrazine (pyz) molecule with respect to the 2D planes.
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