I realize that creating worlds with multiple moons and/or suns is risky
I think I implied this is one of my answers. I suppose that this statement really comes down to opinion, but there's some rationale behind it. The point is this: Multiple stars can be risky; multiple moons generally aren't. So your idea is perfectly fine (I was going to talk about how Jupiter has so many moons, but I realized that you're probably not thinking about a gas giant!).
Is there a way to calculate how often lunar eclipses would occur?
The procedure for doing this wouldn't be any different from the way you would normally calculate when an eclipse would occur: Gather a lot of data and do a lot of calculations. This is actually really, really hard to do, as an explanation here says:
There is no simple formula to calculate what you are interested in. The calculation of eclipses is tedious work requiring many observations or calculations of the positions of the Moon and the Sun. The lunar orbit data must then be extrapolated to find the months when eclipses might occur and then the exact times determined to see where the eclipse will be visible. The eclipse path can then be projected into the Earth's surface to find if a total eclipse will be visible from a given location at a particular time. This work requires a lot of patience and an understanding of the geometry involved and cannot be reduced to a formula.
It kind of stinks, but it's the truth.
. . . especially when all three align?
Well, you really just have to do the calculations for each one, and figure out when they'll all meet up. It's helpful if they're in an orbital resonance, preferably something like $1:2:4$ or $2:6:7$.
By the way, you have to get the sizes and orbital radii of the moons just right. If they appear to small in the sky, you won't see solar eclipses - at least, not total solar eclipses.