How to get just the responsive grid from Bootstrap 3?

Checkout zirafa/bootstrap-grid-only. It contains only the bootstrap grid and responsive utilities that you need (no reset or anything), and simplifies the complexity of working directly with the LESS files.

Go to http://getbootstrap.com/customize/ and toggle just what you want from the BS3 framework and then click "Compile and Download" and you'll get the CSS and JS that you chose.

Open up the CSS and remove all but the grid. They include some normalize stuff too. And you'll need to adjust all the styles on your site to box-sizing: border-box - http://www.w3schools.com/cssref/css3_pr_box-sizing.asp