Tire problem, need some advice

My_third_Z

I got my new rims installed on Monday this week. They are the same size as the previous ones, so I kept the same tires...just moved them from the old to the new. On Wednesday my passenger side rear tire was completely flat, as in 0psi. I put the spare on and went back to the place that mounted them. When they put air in the tire and put soap around it, they found the air was leaking from all around where the tire makes contact with the outside of the rim. They took the tire off, and didn't see any problems with it so they remounted it. Again, it held air just fine for about two days before going back to 0psi.

I currently have the spare on, and am wondering what my next step should be. The tire doesn't seem like it should be the issue, since it was fine on the old rims. The shop didn't notice anything blatantly wrong with the rim either. Could something be wrong with the new rim? They took almost 10 weeks to come in, so I sure hope not. I hate having tires mounted on that rim so many times already, since there's always an opportunity for it to get scratched during the process. What would you guys do next?
Thanks.