if it doesnt show up in a screenshot, it is unlikely to be the graphics card. the screen shot it just a dump of what the GPU is currently outputing. HOWEVER, as it is just happening in games, i would be inclined to believe it is a GPU issue, most probably heat. did you actually check a screenshot to see if it has lines or not?
after the screenshot, if the lines are not there, do you have another monitor you can try?