I've seen heavy curtains on many an audio store, and as the space in my living room could easily look ok with some heavy curtains behind the speakers, I'm wondering how well this idea would work. Anyone ever tried it?
It can help but it is room and system dependent. I have had requests to do a seminar on room treatments because of my rooms at shows and the consistency of good sound I produce (I have had a bad show or two - it happens). I never intend to do one because even similar rooms can be different enough to produce completely different sound.
You could start by using varying thickness of sheets/blankets in the areas and see what happens.
I've used them in the past and they can be very effective - but they won't fix bass problems, and there effectiveness in other frequency ranges varies with several factors.
Some of the things that affect their frequency absorption; the weight, material, 'draped' amount (bunching), how far they are from the wall and what wall material is behind them. They can be very nice for adding a level of flexibility and tuning to the room too, where you can open them to varying degrees changing the draping and exposing whatever is behind them.
In a listening room I built in the country home I built years ago (in a previous life), I first got the bass dealt with by getting the room size right, building in some bass traps and 'staining' the block walls - sealing block with paint hurts the bass absorption. I then used heavy curtains to get higher frequency absorption where I wanted it to get the reverb time reasonable. You could then open or close up some of the curtains to expose the more reflective walls behind them, which had the effect of making the room 'sound' a little larger or smaller. I got so addicted to this tune-ability, that I did something similar in my existing room by using acoustic panels with absorption characteristics similar to those heavy curtains - I move/add/remove them to change from a small smokey bar (my favourite) to a larger space for 'bigger' music. You have heard the small smokey bar and Massey Hall version of this here Erik.
This is well covered right down to all the ugly math in F. Alton Everest's Acoustic Techniques for Home & Studio and The Master Handbook of Acoustics. These are books you really must get IMO - all you've heard in my current room sprung from those pages and my well worn copies are still right here in the bookshelf.