I use a Sapphire HD 6870 and I can assure you that is an excellent card. In 1600x900 (where I play too) will play at the max most games, but in 30-40 fps, so in 1900x1080 you will need to lower a bit the details.
As much as it makes my sad, I would suggest you to buy a stronger card if you can afford it, better one of these (in order form the best to worst): HD 7870, GTX 660, HD 7850, hell even a 560Ti is more suitable for 1080p gaming.
That being said, if you cannot afford on of the above, HD 6870 (or GTX 560, they are practically the same) will serve you well for a good time.