What do the Scriptures give us in answer to what we are to be filling our minds with?
Philippians 4:7-8 (just one quick example)
And the peace of God, which surpasses all comprehension, shall guard your hearts and your minds in Christ Jesus.
Finally, brethren, whatever is true, whatever is honorable, whatever is right, whatever is pure, whatever is lovely, whatever is of good repute, if there is any excellence and if anything worthy of praise, let your mind dwell on these things.
I think that we need to be constantly aware that whatever we subject ourselves to, we are subjecting Christ himself to the same, because he resides in us. If you would be ashamed to be viewing gratuitous violence, whether in movie form or in game form, the acts which denegrate the value of human beings, if Christ were a guest in your home and he were sitting in the room next to you and your engagement in these violent entertainments would prick your conscience, or make you uneasy, I think it is safe to say, you should not be viewing them at all.
This should apply to anything of questionable moral content, whether violence, or sexual content, or the denegration of humanity in any form. We are to guard our hearts, our sensitivities, so that we can be more like Christ, knowing what offends God, what hurts his heart, and also what delights Him. That should be our goal at all times.