GTX 970 Minimum CPU

People will probably need more information than that if you want a direct answer.
AMD or Intel? What socket?

There is no correct answer, there are a lot of factors, the most important being what game you are playing and at what resolution. Lot's of people put up gaming benchmarks for all kinds of games, GPUs, and CPUs. Pick a game you would consider playing and try to find benchmarks done on a 970 with different CPUs. You can probably find people getting over 60 fps in games with a dual core while other games may require a fast quad core or even more.

i7

Too many other factors to consider - game resolution, target fps, type of game/ game genre, graphics settings per game, etc.
Aka there's no easy or cut and dry answer.

Like Alltrac said, go check out some reviews and videos of the 970 in action, and look for the types of games you're looking to run at the settings/resolution you want, and go from there.

One with 4 cores.

I'm still running a first gen i7 with 1070, so i guess you would be fine with the 970.

I have AMD FX-6300 with GTX970 4GB and can say it's not enough for this GPU.

I don't think I can help much, but here's some things I found.

Game Debate recommends pairing a GTX 970 with an i7 4770K, though doesn't seem to say why.

The official Oculus Rift recommended specs include a GTX 970 and an i5 4590.

According to Steam Hardware Survey, nearly 70% of users have a 4-core CPU, and over 50% of user CPUs are 3.0GHz or greater. The most common GPUs appear to be from the GTX 7, 9, and 10 series.

The recommended CPU for some games that list the GTX 970 as recommended:

Deus Ex: Mankind Divided : i7-3770K
Tom Clancy's The Division : i7-3770
Prey (2017) : i7-2600K
Assetto Corsa : "Intel Quad-Core CPU"
Destiny 2 : i5 2400 3.4 GHz or i5 7400 3.5 GHz
Quantum Break : i5 4690, 3.9GHz
Dauntless: : i7 Haswell
Mirror's Edge Catalyst : i7-3770 at 3.4 GHz

So far, it would seem the bare minimum Intel CPU to appropriately pair with a GTX 970 would be an i5 with a base frequency of 3.3 GHz or more.

I feel pretty good with i5 2500k @ 4.2 Ghz + GTX 1060