I disagree that Realtors don't rent property. Most property managers are Realtors and many, many real estate agents earn commissions on a regular basis through rentals.
I do agree that there is very little incentive sometimes to working with tenants. With that said, if you are a qualified tenant, with good references, good income, no credit or criminal issues and no eviction history, make sure to let a Realtor know how well qualified you are and how soon you are looking to move in.
I am SURE that you will find a great Realtor that will work as hard for you as he would for a sale. Because it can be a bit tedious AND there are many UNqualified tenants looking for rentals, I make sure to sort of pre-qualify any prospective tenants before I agree to work with them. When this is done, it is typically a quick and painless process ending with a quick commission and a happy client that will send you referrals and potentially be a buyer one day.
Pam, I do recommend you contact a few property management companies as they might have properties you may not have found listed yet or know other offices that have available rentals they can refer you to.
You can also post what you are looking for on a place like craigslist.com and see what type of replies you get from potential owners of rental property or property managers looking to rent what they have. Some don't like this approach as it can lead to some unwanted emails (spam). I'm a fan of CL.
There is a 'housing wanted' tab under 'housing' on CL. Just another idea if you decide you want to work to find a place on your own.
Best of Luck