This behavior can manifest as “fretting” (hyperventilating), whining, barking, or all of the above, as well as engaging in less creative behaviors--like chewing things, and/or other not quite constructive expressions of angst or agitation. While there are sedatives that the vet may prescribe for extreme cases of anxiety, it may be of some help to look at why a greyhound might exhibit this upsetting behavior.
From the moment he or she came into this world, your greyhound was probably never alone, for even a moment. They are raised in the constant company of their dams and littermates, and while the dam will be separated at some point, the littermates usually remain together. There are often dozens of other pups on the breeder's premises, and they are kept in kennel runs adjacent to one another, where they can be seen, barked at incessantly, and/or goaded into dashing competitions, or display-of-fierceness contests.
Then, in the racing kennel, often the litter remains together, and the larger pack is introduced to them. There, they learn to do everything in concert with their pack/colony, and their handlers, and the atmosphere is quite social.
Even in their crates, they remain in visual contact with their kennelmates and their handlers. Quite often, littermates may spend their entire lives at the same venues, with the same handlers, and remain together until one or more of them is retired.
So, is it any wonder that a newly adopted greyhound, suddenly thrust into what for them is an alien universe, full of strange things and unfamiliar people---and perhaps without the company of other greyhounds, for the first time in his life---might feel some uneasiness?
There can be much more than meets the eye to a greyhound's anxiety. Any number of triggers might induce anxiety in the new adoptee, from the strange new objects and appliances in the home, to the new smells, sights and sounds of the neighborhood, to any of the many changes in his established and ingrained routine, to which he/she must now learn to adapt.
The most overlooked of these triggers being, that the greyhound has no idea what he did wrong to have suddenly been picked up and plopped down into this entirely new, and (often) intimidating situation. There is a blind spot among some adopters, which can fail to perceive even the possibility that the greyhound may have been perfectly happy with things as they were, as a racing athlete, one among many---a pack member.
Contrary to popular greyhound mythology, the vast majority of racing greyhounds, are quite content and fulfilled doing what it is that they have been bred to do, within a colony of their peers. Working dogs are generally that way. Most relish and thrive on their work, and the physical and mental stimulation it provides.
Greyhounds prosper with routine, punctuality and repetition. They blossom when they are as free of all stresses as we can make them. But they often have some reservations about novelty. They are used to regimentation and predictability, and their whole lives have revolved around the narrower confines of the breeding, raising, training and racing environment, as opposed to the brave new world of the adopter's home, social outlets, and leisure time activities, in which the dog may now be included. Regardless, he no longer has the outlet of training and racing to pleasantly fatigue himself, and to relieve pent up stress---a very important factor to be aware of.
The new, retired adoptee was likely already bonded to one or more of his/her handlers, and often, to one or more of their kennelmates---who are now, suddenly, gone. It's a huge void to fill for most of them. This bonding, by the way, generally happens over a period of time, where the greyhound learns who, in their circle, can be relied upon and trusted. Just because a newly adopted greyhound may resign himself to the fact that you are his new human, and even be amenable to it, doesn't mean that you have bonded with him--or he with you. That may or may not happen, with time, depending upon your individual greyhound’s adaptability---and your own.
The point is, of course, that separation anxiety can be more of an "I simply can't deal with being alone, and I miss my job and my friends" anxiety---especially for the new adoptee.
Smothering the dog with toys, treats and attention won't usually be a panacea for the anxious, newly re-homed greyhound. That elusive panacea is more likely to be routine, punctuality, stress reduction in the home environment, physically and mentally engaging the dog in stimulating, healthy activities--and time--time for the greyhound to learn to trust, to rely upon, and then to eventually bond with their new person(s).