These two follow-up approaches have very different goals!
Univariate ANOVAs (as follow-ups to MANOVA) aim at checking which individual variables (as opposed to all variables together) differ between groups.
Linear Discriminant Analysis, LDA, (as a follow-up to MANOVA) aims at checking which linear combination of individual variables leads to maximal group separability and at interpreting this linear combination.
This question asked about one-way MANOVA with only a single factor, but see here for the [more complicated] case of factorial MANOVA: How to follow up a factorial MANOVA with discriminant analysis?
So e.g. if your individual variables are weight and height, then with univariate ANOVAs you can test if weight and height, separately, differ between groups. With LDA you can find out that the best group separability is given by, say, 2*weight+3*height. Then you can try to interpret this linear combination.
So the choice between these two follow-up approaches entirely depends on what you want to test.
Two further remarks
First, if you are "interested in how the three groups influence every dependent variable" (i.e. individual DVs are of primary interest), then you should arguably not run MANOVA at all, but go straight to univariate ANOVAs! Correct for multiple comparisons (note that Bonferroni is very conservative, you might prefer to control false discover rate instead; but see comment below for another opinion), but proceed with univariate tests. After all, nine DVs are not a lot. If, instead, you are interested in whether groups differed at all (and maybe in what respect they differed the most) but do not care so much about individual DVs, then you should use MANOVA. It all depends on your research hypothesis.
Second, it sounds though as if you might have no pre-specified hypothesis about which DVs should be influenced by group, and what exactly this influence should be. Instead, you probably have a bunch of data that you wish to explore. It is a valid wish, but it means that you are doing exploratory analysis. And in this case my best advice would be: plot the data and look at it!
You can plot a number of things. I would plot distributions of each DV for each of the three groups (i.e. nine plots; can be density plots or box plots). I would also run linear discriminant analysis (which is intimately related to MANOVA, see e.g. my answer here), project the data onto the first two discriminant axes and plot all your data as a scatter plot with different groups marked in different colours. You can project original DVs onto the same scatter plot, obtaining a biplot (here is a nice example done with PCA, but one can make a similar one with LDA too).