@Mahesh P.
We are experiencing some of the same issues that this family is.
Our family has not lived in the US for the past 25 years. My son is set to go to college so I decided it was time to take a job in the US to support him while is in school. I have no landlord recommendations in the US and my overseas housing was provided by my company, so they have no idea that I even lived in the place (it is all registered to my company). My situation may be a little easier because my employers have always been US companies and I can provide verification for everything. I have US credit cards as well as International cards, so my credit is checkable and just above average. Another twist in the story is that my wife is not a US citizen and at this time has no Visa. We are getting her a green card, but it has not come through just yet.
What do you think my odds are of being able to rent an apartment or house?
My advice to you would be to check and double check. If they are US citizens then there should be information about them in the normal systems (credit, bank, drivers license, etc.). If it comes back positive, then your fears are probably misplaced. But, if any part of it does not check out or is negative, then do not rent to them, you are setting yourself up for trouble.
We do not actually have any issues because we are buying straight away and do not need to rent.