In case you hadn't heard, Samsung is making a little extra effort to secure the Galaxy S4 via tougher software kernels, which aren't susceptible to some of the more common root methods from previous Galaxy models. But where there's a will, there's a way, and noted Android developer/modder Chainfire has found a way around the security on the stock kernel for the Galaxy S4 i9505 - that's the Qualcomm Snapdragon-powered LTE variant, and the model sold for AT&T and T-Mobile in the United States.
This root exploit works via a newly-modified version of the CF-Auto-Root tool and the popular SuperSU permission manager app. You'll need a Windows computer and the latest version of Samsung's Odin flashing software to apply the MD5 file. The root method should work just fine with T-Mobile's Galaxy S4, and AT&T's model as well, assuming that you've unlocked the bootloader via Dan Rosenberg's method. Some users have reported that the modified CF-Auto-Root works on Sprint models (SPH-L720), but this has not been verified, and caution is advised for the present.
Unfortunately, this root exploit will not work on the international HSPA Exynos-powered Galaxy S4 (i9500). Chainfire mentions on Google+ that he's working on a root solution, but since he doesn't have physical access to one of these models, it could take some time.