Will I have to do a checkout ride at a rental place if I'm already familiar with the aircraft type?
Generally they will make you do a checkout with their own instructor, at least this is the situation I have always encountered when renting from various places. Generally speaking, when you go to a new rental place they will audit your log book, put you in their system and make you fly with one of their instructors for a few reasons:


While your license was issued, presumably legitimately and your log book may be full of hours that are presumed to have occurred they still want to make sure you can do what you say you can. 
Generally their insurance will require it. At least this is what I have always been told by FBO's/Flight Schools. Both of the places I fly out of, even require a recurrent 90 day checkup with an instructor to be able to keep renting (no matter how much you fly). 


One thing to note is that none of these are FAA regulations, so you are legal to fly the PA-28 as long as you have your bi-annual and TO/LD requirements filled. The rental shops can how ever impose other rules as they see fit (mine does not allow flights to grass strips with the rental planes). 

Note: some rental shops may also have differing gap policies on their insurance. It's worth asking about this up front and seeing if you need to carry any gap insurance.