I see this question all over the internet, and I'd like to answer it once and for all. First of all, this is my system:
Windows 10 Home 64-bit
Intel(R) Core(TM) i5-6402P CPU @ 2.80GHz (4 CPUs)
Memory: 24gb
NVIDIA GeForce GTX 1070 8gb
I just purchased a 4k monitor, and it runs smoothly with this setup. Like everyone else, I wasn't sure if the graphics card could handle it. Well it does, easily. Granted, you won't be able to run everything on Ultra at 60 FPS, but turning down to High settings is well worth it. The picture is crisp, the colors are vibrant, and it feels like looking through a window.
My theory is that the extra RAM helps my computer keep up. So if you have a 1070, you might want to invest in the RAM as well. Today I played Elite: Dangerous and Battleground on Ultra (the latter of which I decided to turn down to High when I got a frame rate drop). I couldn't be happier with my choice of buying this monitor.
Point being... GO FOR IT!