The R270 will work with the U3415W. The only thing to be aware of is that if you're using this monitor for gaming (with the 270) that the high resolution might cause games to have a very low fps (unless details are turned down). I have seen some benchmarks with the r295 and even that card has a hard time on newer games (full res and details). So for gaming a new GPU might be recommended too (at least eventually).
Also, on a side note you will want to hook this monitor up using display port, for some reason I want to say the 270 does not have the newer HDMI 2.0.
Sorry for the delay in reply, I would suggest looking at this thread here on Video Cards that have been tested by other members. From the looks of it a 970 and up to a 1070/1080 GTX will work. I would recommend looking at video card reviews but from what I have seen a 1070 would be the minimum I would try with this resolution, unless you wanted to turn down other settings. Again though, this is only if you will be using the system to game.
In addition, this thread I linked above and others on here will have various suggestions on power supplies and settings to get the cards to work. Aside from this, I would recommend to update the system BIOS.
I hope this helps.