Please comment on your experience with Plantation shutters. Yes they look great,

but do they act as good sound diffusers over windows? Has anyone noticed they improved the sound in a room with windows? I am thinking they may do a good job when tilted, but not fully closed. Perhaps even better than 2 inch wide standard shades?
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You are dealing with two issues with plantation shutters, 1) Diffusion off the blades which will be infinitely variable over 180 degrees (you could spend a lot of time trying to dial that in alone) and 2) First reflections off of the vertical glass behind them which will not diffuse at all. Neither will absorb any sound which IMHO is  a much more important issue when dealing with first reflections.

 I think any control of the sound waves that you might obtain by varying the pitch of the blades will be lost because of the reflecting nature of the glass immediately behind them. Heavy toe in and/or heavy drapes which you can open close when listening to music would be a much better resolution.

Now if the plantation shutters/windows are immediately behind the speakers or the listening chair, that would be an entirely different situation and I think they might work fairly well in place of more audiophile type of treatments.

Shutters will do little to nothing to help I'm afraid. Heavy drapes is an option but if you have the chance attaching something such as ASC sound planks to the window can work wonders. You can use Velcro to attach them so they can be removed to open the window when not needed. I am sure other similar items would work as well. Also consider the Marigo labs tuning dots for a very low profile way to damp the window

Marigo dots will do very little/nothing to help with the window.....lows travel through them like they were open.....highs get reflected right back at ya.  My house has great walls of glass windows....I use drapes recommended by the decorator....they look nice but do some, but little to perfect the room acoustically. Sometimes you go for the look, sometimes for the sound.....rarely can they be combined.
I have a double window (two completely seperate isolated panes mounted with a small offset between them to reduce chances of resonance in the window) with shutters in between as well. The inner window then has sound planks (three on one pane, two and a SR ART resonator and HFTs on the other) and also the large (40mm) Marigo dots on each pane

Each part helps but the main conclusion is you can (and probably should) treat a window like any other wall but obviously the end result is not going to win any interior design prizes ...

In a prior installation I lived with acres of glass and light treatment with the SR ART system and while no where near as acoustically well controlled as my current set-up it was very listenable

ps there is a fairly long track record of Marigo dots working well on exposed glass -- it can’t hurt give Ron Heydrich a call he’s a great guy and may have some good ideaas
Certainly closed or near closed wood plantation shutters is a better surface than glass for sound? Not great, but better.....
Well the only practice difference between shutters and glass, or any flat surfaced wall, is the slight curved nature of the blades will give a small but uniform  bit of diffusion. It won't diminish it - how that might sound I don't know. You are right though, it will be better than uncovered glass. If your speakers aren't hot on axis, toe in may still be the best solution
I plan on toe in for sure. My wife will not allow drapes, but she loves these expensive plantation shutters. No magic sound properties it seems! 
In my limited expertise, I think the bandwidth over which shutters diffuse is going to be fairly narrow.
sounds like the decision has already been made
Install drapes that hang outside of/over the shutters.
Keep the drapes open except when you want to listen.
Tell your wife it is a necessary energy saving measure for the cold Minnesota winters.
Good luck with that.
Mitch2, great response! I will try this approach and idea.
Mitch's response makes me realize that the shutters could be installed on top of/over the acoustic window treatments I described (sound planks et al), so you could treat the window and have the look your wife is after. As others have noted the shutters themselves are acoustically transparent so there's nothing to stop you treating the surface behind them and getting the look you need. Heavy drapes will actually be very expensive so wonder if you should try something else before you spring for that?

My room has three large windows on the left wall with plantation style shutters.  The speakers are designed for a 45 degree toe in but I prefer them at 40 degrees.  Therefore, sidewall reflections are not a major factor.  The toe angle intersects in front of my listening position. The speakers also have a side firing tweeter which on the left side is 23 inches from the window so there is some direct sound involved from this.

I prefer the sound with shutters on the first two windows closest to the speaker pointing downwards and the third window shutter pointing upwards.  In this configuration the imaging is more precise and the bass better defined.  The right side speaker intersects on axis with the third window.   With the shutters open to the horizontal position, the sound does get brighter and less focused.  Fully closed in this orientation gives a more intimate denser sound.

Reversing the orientation described above results in larger sound with bass that is less defined and a more forward presentation.

I'm satisfied with the sound and the opportunity to make adjustments with predictable results.  In my environment of AZ, I need to block out the morning sun as much as possible.  Buy the shutters.  I think you will be satisfied.

I use those shutters- no problem. Happy Listening!
They will obviously reflect sound differently than bare windows. If that is a good or bad thing will depend case by case. Its the acoustics of the room as a whole that matters, not any single part, though any of those might be tweaked as needed to get desired results.