Random BSOD on brand new computer

I'm having random BSODs on my brand new computer: I5-760, 8GB Ram Kingmax (2x4G dual core), video Radeon 6870 1GB, Windows 7 x64 fresh install.

I used BlueScreenView but looks like the error code is different every time, so I am a bit lost. I attached all the files required for posting BSODs and I am kindly ask for help.
Thank you.

P.S. I actually forgot how a BSOD looks like, my old PC with Windows XP didn't give not a single one in 7 years of heavy utilization. I suddenly feel like in old ages of Windows 95/98

Remove any Eset Node32/Smart Security Suite and replace it with MSE

I removed Node32/Smart Security Suite and replaced it with MSE, but got another 2 BSODs after that (one of them before I rebooted to finish installation of the new AV and second one this morning).
I attached last 2 dumps.

Update your HD6870 drivers
Previous ATI Catalyst

Updated video card drivers to 10.10 (actually I downgraded them, since the installer reported everything as up to date), but nothing is changed, BSODs does not want to go away.
I attached last 5 bsod reports.

The most recent dumps points to Ntfs.sys related to your hard disk, most probably caused by disk/system file corruption
Run Disk Check and System File Checker

I run both disk check and system file checker, first one didn't reported any error, while second repaired some files.
However, I got another bsod after I did that, and couple other before I managed to do the checks (see attached). Last couple crashes seems to be memory related, so I also run memory test from windows, but didn't find any problem.

I'll remove one of the memory sticks to see if it still happening. Any other ideas?

Later edit: after I got 3 bsods in a short amount of time, I put Memtest86 on a stick, booted from it and tested each memory, one at a time. First one passed all test, while second got tons of errors. Apparently I have found the cause, I'll keep you informed if this "cured" my problem.