RB,
Glyphosate (Round-up) shouldn't hurt the tree at all if applied correctly.
The method others suggested will also work, but I actually prefer the chemical method as it doesn't interfere with the oxygen transfer to the soil. Just a personal preference, though, and not backed up by anything but solid scientific conjecture.
Only spray on enough Round-up to lightly wet the grass. No runoff, in other words. In order for glyphosate to kill plant material, it must be absorbed through the leaves.* Glyphosate is broken down incredibly quickly by soil bacteria into very harmless components, so what little does reach your soil won't do any damage anyway.
It can take up to two weeks for the grass to turn brown and die, so don't worry about it when it doesn't start dying right away. Once the grass is dead, it's your choice to remove it or not. Personally, I would just scalp it with a string trimmer annd throw the mulch right on top of it. To my mind, digging the grass would cause more injury to the tree roots than anything else.
* Not completely true, because dumping it on the ground would cause things like salt injuries, etc. to roots, but for all practical purposes...
p.s. Thanks for the rep, BTW...