Asus HD 6870 DirectCU AMD Radeon or PSU problem

Sounds about right. Over-spec'ed, low quality power supplies are like that. They also have a very bad habit of dying when stressed and taking other components with them. Can make it rather expensive having to replace the motherboard and possibly other parts in addition to the power supply.

A good read on what to look for in a power supply,

Power Supply Information and Selection - Tech Support Forum

and what to avoid,

The Bargain Basement Power Supply Roundup Review

Yes now it can still be the gfx card or the psu.
I suggest you to run Furmark at a low resolution to stress your card to 100% but not your CPU. If your CPU usage stays low and card usage close to the top, and it doesn't crash, than it is your PSU.

You can see yourself here.
I'd buy a 500W Gigabyte Superb/500W coolink. If you want to go pro you shall go with the 500W Corsair CX.