I have a similar setup, my miner is mounted on a wooden board and is very close to where I sleep in a small room. I very much do not like noise and generally tolerate much less than others but I've finally gotten my system to a point where heat is more of a concern than noise. Before obtaining this miner all of my computers were solid-state. I only attached HDDs using a USB harddrive enclosure so I could switch them off when not using them. I've not had experience running a fan in my room 24-7 for nearly 5 years (I used to have an ordinary computer tower, not particularly noisy, but running it 24-7 drove me mad so I got rid of it and replaced it with a solid-state computer).
You can find pictures of my rig here
I've attached Zalman VF3000A coolers to my Sapphire HD5850 Xtreme cards. The coolers are designed for reference 58xx cards and my Xtreme cards are built much more like some of the 6xxx family so I had some difficulty mounting the coolers properly. The fans on the coolers are 3-pin whereas the fans on the cards are 4-pin so I have to run the fans from the motherboard (if you want the cards to be able to control their fans then be aware of this if you decide to go down my route). By using the enclosed Fanmates and turning them down to the minimum (which I think is about 50% fan speed) the coolers are pretty quiet (a fair bit quieter than most PSUs I'd say).
I replaced my 650W OCZ PSU with an 850W Scythe Couriki 2 PSU. This isn't the most efficient or feature filled PSU but it's not bad and it's the best I know of for silence. This power supply is a fair bit quieter than my two Zalman ZF3000A fans at minimum speed so if your PSU is giving you trouble then I recommend this. 3 5870s should be no bother for it (My 2x5850 system is currently drawing 304W at the wall).
The CPU fan is actually a little quieter than the PSU because it's running at minimum speed (It is rare that CPU usage exceeds 5% so the fan is never asked to go faster). However, if I manually turn this fan up to full speed it completely drowns out the rest of the system so make sure these fans are at minimum speed.
I'm using a HDD and it clicks away regularly because I'm running bitcoind but I actually quite like the sound of the HDD so I'm not bothered. You may want to run off of a USB or in RAM to eliminate this sound.
Last, but not least, it is a very good idea to undervolt your cards. I took mine down by a full 0.1V (from 1.0875V to 0.9875V). This reduced my stable hash rate from 830 MH/s to 724 MH/s, and this may seem like a lot to lose, but the drop in noise, heat, and power consumption is impressive and I highly recommend this. The MH/J of my system is now very high (2.38 MH/J); I was barely over 2 before.
With all of this I can finally tolerate the noise level. It would be nice if it was quieter but I can't think of a way of making this happen without installing water cooling, and I would have to be careful to get a quiet reservoir (I believe my system is quieter than some fully water-cooled systems already).
I'd be interested to learn how you fare with voltages this low or lower (I currently have little information to go on and am trying to improve my MH/J).