TxBecky
How can I get grass to grow under my live oak trees?

I live just outside of Austin Texas and have lots of these trees. The problem may be they type of grass that you are trying to use and the depth of the soil.
These trees send roots out just below the surface and sometimes the roots run on the surface of the soil. You will need to add soil to get enough drainage so that water is not blocked by these roots. There is a problem with a fungus that will attack patches that are too moist on warm days and the grass will die. Lots of folks here like the deepness of the St. Augustine but that grass has a hard time with moisture pools. Fescue is the best type of grass for areas that are hot and humid. It is also the xeriscaping choice.
Just some ideas.

Trees take in huge amounts of water. If the area where the tree is well drained and in a southerly aspect, then growing grass will be particularly hard as you get closer to the base of the tree. It may be than rather than trying to get some bare struggling grass patch, why not try an alternative? A bark mixture on the ground may be a good alternative. Alternatively or in conjunction with the bark, why not plant some winter/spring bulbs which will grow and flower while the oak is in its resting period.