Here is an external link to one of many resources that can help provide perspective on your question:
It sounds like you are looking for a MANOVA. The question of complicated is more of a relative one. If you look at the resource above, you will see that:
- A two-way ANOVA has two independent variables (which matches your case). That said, a two-way ANOVA still has only one dependent variable, which doesn't seem to fit the case that you described. Now, perhaps the two dependent variables are not actually dependent variables, and are levels. If one was able to recast them like this, then you might be able to fit things into a two-way ANOVA. That said, it doesn't sound like this is the best approach.
Based on this, we can then conclude that your case may better be cast as a MANOVA. A MANOVA fits the scenario when you have several independent variables and several dependent variables (which you described). If you have never done MANOVA before, and you happen to use R, then you might have a look at the excellent resources provided by QuickR covering MANOVA's.
As they show under the simple MANOVA example (which debunks the notion of complexity in my mind), if you have more than one "dependent (outcome) variable, you can test them simultaneously using a multivariate analysis of variance (MANOVA)."
In your case, we could let Y be a matrix whose columns are the dependent variables. Thus, with 3 dependent variables, and 2 indepedent, you might have the following R code:
Y <- cbind(y1,y2,y3)
fit <- manova(Y ~ A*B)
For information on testing, you can look online in a variety of places. The link shown at QuickR does provide a brief description.