There really isn't a straightforward generalization to Cramer's V, in the same way that Pearson's $\phi$ isn't defined for $2 x 2 .... 2$ tables. This is because there are multiple null models to be selecting from, so a single collapsed relationship statistic is not meaningful.
As an example, say you have a large $\chi^2$ value in your association test. In a $2 x 2$ case this is simple, because the rescaling of the $\chi^2$ to a correlation is still meaningful: one unit of increase has a correlation of $\phi$ with the second variable. However, for a $2 x 2 x 2$ model, a large/significant $\chi^2$ could mean that there are many possible relationships present (V1 with V2, V1 with V3, V2 with V3, or V1 with V2 with V3). So a single number would not capture where this co-relationship is occurring, and hence a single correlation statistic would just be ambiguous. The same principle applies to Cramer's V, where higher dimensional tables are not easily collapsible to single number relationships.
Overall goodness of fit measures that are analogous to $R^2$ in linear regression are possible and several have been proposed (e.g., Pseudo R squared formula for GLMs), though these are not that easily done by hand as they rely on computing log-likelihood/deviation values.