What method to use to analyse 3 groups of independent variables? I’m trying to find a better way to analyse my data.
Unfortunately my limited background in math does not allow me to freely make a confident choice. I’ve already done an ANOVA, the method mostly used in our lab. As it often happens with statistics in non-statistical fields, it was chosen mainly based on the criterion, “Everybody uses it,” rather than, “It is the best way to do it.” This approach is leaving me in a state of an itch. I understand, ANOVA indeed could be the best choice in my analysis, but again, I am not capable enough to make comparisons between different methods.
I would appreciate the opinions and advice of a statistic-minded community.

Short overview of my project.
I’ll try to make it as less specific as possible.
Let’s imagine a brick wall.
It has standard bricks and mortar in it. The wall has a measurable quality, let’s say a ratio between how long it needs to be vibrated until it collapses and the amplitude of the vibration. Call it G.
G depends on both wall components—bricks and mortar.
Mortar has two components on its own, like cement and sand.
What I am trying to find is what relationship my special shaped bricks may play on overall G in different conditions.

Independent variables.
I divided my independent variables into  3 groups—A, B & C.
Group A
Changes  in the mortar composition and has following arrangements:
A0—nothing changed
A1—reduced cement from the mortar
A2—increased cement in the mortar
A3—cement replaced to the new type of the cement known to affect G of the wall
A4—changed shape of the sand particles
Group B
Changes in the shape of the bricks
B0—normal wall with all normal bricks
B1—added bricks with Shape 1 into wall
B2—added bricks with Shape 2 into wall
B3—added bricks with Shape 1 and 2 together
Group C
Layer of paint on the wall
C0—no paint
C1—thick layer
C2—thin layer

For every combination of A, B and C I have 20 to 50 readouts of G.
What would the best method to employ be to say, e.g., "Combination of two shapes of the bricks together is the main factor in keeping G at a certain level while mortar has changed and paint is applied?"
I was thinking to make a multiple regression analysis, as I also have quantitative data of A0, A1 and A2, but not A3 and A4.
Making multiple ANOVA comparisons in this case feels the same as making multiple $t$-tests.
I would appreciate any thoughts.

Edit
Here is how my creative graphing looks like.
Orange belt in the middle is a “zone of insignificance” from my ANOVA (compare mean of each column with mean of control column (A0B0C0).
What I wanted to say is something along this line:
“The effect of C2 on the wall is attenuated while B3 is present despite disturbance of the mortar with A1-A4"
 A: A three-way ANOVa with factors $A \times B \times C$ sounds pretty reasonable to me. Then you can see whether or not A, B, or C or any interactive combination thereof influences G.
The downside of this procedure is that if you get significant effects you are only little wiser. You still need to find out which of the levels drives the effect. I think the prime advise to give you is: plot your data (I would start with a plot of G on the y-axis, A on the x-axis, B as different lines, and different plots for the three levels of C).
You can then use the significant results and the results from the visul inspection of the plots to run follow-up analyses on the significant effects (post-hoc test or simple effects analyses).
One thing you haven't mentioned and I took for granted: Is your dv, G, interval scaled? This is necessary.
A: Testing linear contrasts might be preferable to simple omnibus testing of differences with respect to some of your factors. I'm still not clear on what level of measurement you're working with, but group C at the very least seems like one ordinal variable to me: "Amount of paint added (with levels 0, 1, and 2, unless you have more precise measurements [in which case it might be continuous data])," as I said in my comment. I don't know what theories you want to test, but it seems reasonable to me to test a linear theory about the effect of paint, such that whatever effect it has would be proportional (directly, inversely, or whatever) to the amount of paint added. A basic ANOVA on one variable with three levels (as I've framed it; hopefully this fits what you're actually working with) tests for significant overall differences among these three levels. Therefore it's less powerful than a $t$-contrast (or $F$-contrast if you have more than one linear theory to test) for identifying a steady trend of linear increase, if that's the particular theory you set the test to evaluate.
