I will be using BF4 since it is the most graphically demanding game you listed.
now, to answer you question: that depends, judging by the following image the GTX 970 can run a single 4k monitor at about 26-37fps with high preset
now, this is a bit of an iffy comparison but I may be able to assume that performance here is directly related to pixels introduced due to the fact 3x 1080p monitors is just about 6220800 pixels adn a 4k monitor has 8294400(first off this is an ungodly amount of pixels good lord
) and 8294400/6220800 is about 4/3 it can be somewhat assumed that a 3x 1080p setup would deliver about 4/3 the performance of a 4k monitor. which in the case of the statistics above is about 35-50fps at a high preset. This is completely theoretical though so due to the spread of the monitors it may be slower/more "sttutery". With this I believe it is safe to assume you would get about 30-50 fps in bf4 at the high preset off of one GTX 670.
While I understand that what I just did might be a bit unrealistic and I understand it does not scale perfectly I believe those numbers will give you a rough estimate as to where the performance will hit.
Be that as it were, one gtx 970 will be able to run 3 1080p monitors at high settings in bf4... that being said, I think you should deffinetely consider getting a second card if you plan on gaming at that resolution, it will be much more enjoyable.
note: I used this method because I was unable to find any benchmarks for 3x 1080p in bf4 on the gtx 970, at least from the short time I looked.