If you simply want six drives out of the way and accessible then I am inclined to use a large case such as the Lian-Li and be done with it. You have all the needed SATA ports on your board, so I see no reason to make things more complicated.
Get a nice large Full Tower case with as many drive bays as you need with some extra for upgrades later on. Also keep in mind for later expansion you can get mounting brackets to turn your 5.25" bays into 3.5" bays for hard drives.
I have a mid-tower case housing my HTPC with seven drives in it. It has four regular HDD bays and three optical bays. I tossed in a 3x 5.25" to 3.5" mounting kit and have all slots filled up. 4x 2TB and 1x 3TB, and an old 80GB IDE drive with the OS on it. No need for an optical drive in this system once windows was installed.
Next time I need to add a drive I will have to move up to a nice full-tower case and get a SATA card since the 6x ports I have are all being used. If I had any idea how fast the system would grow I would have gotten a full-tower to begin with.
As far as particular cases go, have a look here, use the advanced options on the left an put in however many "Internal 3.5" Drive Bays" as you feel are necessary and find one you like. I would say at least 8 plus whatever 5.25" bays you could convert later on. Cases are a bit personal, so you need to look for what appeals to you and fits your budget. If you have any questions about any particular case post back and we will help.
Ultimate Edition Linux
AMD FX 8350
Asus Sabertooth 990FX R2.0
24GB GSkill/Corsair 1866MHz
EVGA GTX 970 SSC/ GTX 650 (PhysX)
SSD - OCZ V4 128GB/Kingston 120GB
HDD - 3TB/2x 750GB/500GB
CM HAF 932 Red
CM Hyper 212 EVO w/2x 120mm Red
Logitech X540 5.1
2X Acer 23" LED - Surround: 3840x1080
Microsoft Sidewinder x4 KB
"The significant problems we face cannot be solved at the same level of thinking we were at when we created them."
- Albert Einstein