I just purchased a 10 x 10 gazebo and plan on putting it on my concrete patio. How should I anchor it to the patio without having to drill into the concrete?

There are huge U shaped bolts that you can put into the ground where you use the U shape to attach heavy wire rope to. This configuration is similar to the tie downs that are used for mobile homes.

You could add cement blocks or bags of sand to the legs. Adding unsightly weights is about all you can do if proper anchoring is not possible. Why are you not wanting to drill the concrete?

If the gazebo is heavy enough, you will not need anchors. If it is made of wood, then you can use L-brackets, bolt to wood then, glue to cement using a high strength epoxy. The recommended procedure is to drill to cement using expansion bolts, especially in high wind areas.