Best Value Video Card for 4k monitors and improved rendering

I also second RoninEdits. You need to also upgrade the CPU in order to justify the cost of that GTX 1070. Otherwise, the GPU goes largely to waste because it must sit idle waiting for the CPU to catch up. You see, not only is your CPU several generations old now, but it is only a quad-core with hyperthreading CPU. As such, it is slower than current-generation Skylake and Kaby Lake CPUs which use "lesser" CPU sockets and chipsets. Only get a new GPU if your current GPU is no longer supported for MPE GPU acceleration or after you upgrade the CPU.

My suggestion would be first to upgrade the CPU.  You do not have much choice with that 2011 socket but I did see an i7-4960X used for only $300.  You would get much better performance over a GPU upgrade.  When you get some CPU horsepower then think about a GPU upgrade.

Ronin you beat me again while I took my lunch break

improved rendering is often linked to the cpu performance. finding a used cpu with more cores, but still high clock speeds, that will work with your motherboard may be a good cost effective cpu upgrade. the i7-4930k is one example that might be easier to find used for a decent price. it has 6 cores and could be overclocked for even more performance. the website for your gigabyte motherboard will list which cpu's are compatible.

a newer video card should be able to run three 4k monitors. however the gtx 1070 is way faster than what your current 4 core cpu would normally benefit from, and even a 6 core would likely not need that much gpu performance unless you are doing lots of gpu fx, and/or very intensive ones like neat video denoise. a gtx 1060 6gb would typically be a better match to a 6 core cpu.