I think it depends to some degree on the degree of departure from normality (check the skewness and kurtosis values). Very minor departures won't affect Pearson's & ANOVA too badly. More serious ones will.

If the departures from normality in your 3 non-normal variables are quite serious, I'd suggest using non-parametric analyses for all of them - based on the argument that the non-parametric analyses are still valid for the normally distributed variables (albeit with some data reduction involved), while the parametric analyes would likely NOT be valid for the non-normally distributed variables. It'd certainly be less convoluted than doing parametric tests for the relationships between some pairs of variables and non-parametric tests for other pairwise comparisons.

Just a sidenote: on the off chance your 6 variables represent measurements of some kind of the same subjects, you would need to use repeated measures analyses rather than Kruskal-Wallis etc.

Hope that helps!