Although separation anxiety can cause digging episodes, its influence on this behavior is much less than with other problem behaviors. Instead, sheer boredom and/or instinctive behavior (especially a breed-related "drive" to dig) are the two common states of mind that compel a dog to dig.
Dogs with nothing else to do might opt for yard excavation just to help pass the time or to use up extra energy. The urge to break out of confinement and roam the neighborhood can also compel a dog to start digging. Finally, as you might have already experienced, many dogs like to bury personal items such as bones or toys for exhumation at a later date. Such instinctive behavior, though aggravating, can hardly be considered abnormal, and is difficult to totally eliminate.
Increasing your dog's daily dose of exercise could be just what the doctor ordered to help resolve its boredom and release any pent-up energy. Diverting the attention of a chronic digger is another plausible treatment approach. For instance, some troublesome cases have responded very well to the addition of another canine playmate. Rawhide bones and other chewing devices can also be used as attention-grabbers, but only if they don't end up underground themselves.
If most of the digging occurs at night, overnight confinement to the garage might be the answer to spare your yard from the ravages of claws. Finally, if you haven't already done it, neutering can sometimes help snuff out the strong urge to dig in those dogs wanting to escape the yard and roam the neighborhood.