PC Black screening then restarting while gaming

Well your Psu seems fine and up to the job of driving the new 970 card you have.

So next step is to check what happens when you play a game that puts load on the card.
Download a temp monitor for your card and keep an eye on it when playing a game.
You should find out what the thermal limit is of the Gpu under moderate or heavy load depending on the games you are playing, the resolution playing at and the FX you have turned on in the game.

If the card reaches the stated thermal limit, you will first notice a drop in frame rates within a game.
That can be followed by the actual video driver terminating and dropping back to to windows desktop in order to let the Gpu cool down.

Your other causes,in respect to the system restarting can also be a thermal limit on the cpu or the I5 you have.
Depending on your cooler you have fitted on your I5. if you are using an Intel stock cooler the temps will be quiet high.
In the bios of the motherboard you should have thermal protection options.
where you can set the value in degrees C/F when the system forces it`s self to shut down.
As a safety feature to protect burn out of your Cpu.
Again you should monitor the temp of your I5 cpu and note at what temperature point the system restarts at.
Then look at the values set in the bios.
If the temps seem high then it can be a sign that the stock cooler on the I5 is not fitted right.
Or you can change the value in the bios that triggers a thermal shut down of your system if the current value set is too low.
If you have not over clocked the cpu or the system in anyway. Then it is likely a thermal problem.

I would also check that the bios has set up your memory right and selected the right speed for it to run at and that it has correctly set the right memory timing settings that can be found on the sticker of your memory, along with the correct voltage settings for the memory modules.

Consider if you have room in the case, if you have not already done so of adding a push pull fan configuration to help with cooling the internals of the case.

It seems to me though that you have a thermal problem, At a guess maybe with your I5 cpu and it`s cooler.
Monitoring the Cpu temps and Gpu temps will give you the answer to the restart of your system though.