Wow. A lot of negativity from members. If the car is seven years old and has 75,000 miles, you are covered under the 10yr/100k mile warranty; simple. If I were you, I would go get the car and not return unless you get a loaner vehicle. The dealership does not have to give you one but since you're under warranty, it's common practice. I don't know how old you are but it definitely plays a role when it comes to dealership courtesy. The younger you are, the less willing they are to go the extra mile.
Go to the dealership and demand they give you a loaner. If not, demand they put your car back together, if applicable, and go to another dealership with loaners available.
Based on what Jorge is saying, Hyundai is attempting to fix the issue. He seems upset because it is taking a fair amount of time and they didn't offer him a loaner. He provides no information as to what the issue is, what the transmission was doing that lead him to believe there was a problem, not does he seem to have any understanding of how a vehicle works or what it takes to repair.
But based on what he is saying they are honoring the warranty, as written. They are attempting to fix the vehicle at no charge to him = honoring warranty.
No place in the warranty does it state he is entitled to a loaner car, or any type of compensation for the inconvenience, nor does it state that they can snap their fingers and magically fix it in a day or two.
Transmissions are complicated and troubleshooting, diagnosing, and repairing take time, and that is if there are no back ordered parts or supply issues.
3 weeks is not that long, I had a failed transmission on a 2005 Chrysler 300cSRT that was warranty covered and it to 9 weeks to get the car back. I was still within the bumper to bumper so I did get a loaner per THAT warranty by Chrysler, had I been out of the bumper to bumper and into the drivetrain warranty I would not have been given a loaner.
The issue in this thread is plain and simple, the OP doesn't have knowledge of how his vehicle works, does not understand what it takes to diagnose and repair the issue and wants an instant repair and for Hyundai to compensate him for the inconvenience of not having a vehicle, and no where within the warranty by Hyundai or any other manufacturer for that matter is your personal inconvenience covered. Nor does it state they will replace the part with a new part, the warranty entitles the customer to a repair done at no fee to the customer, plain and simple.
Hyundai is doing what they need to do and are obligated to do, testing and diagnosing the vehicle at no charge to the customer in an attempt to resolve the issue, Jorge seems to dense to grasp this, sorry if you find that negative or harsh, but that is the truth of the matter as I read it in this thread.