First of all, I'm so sorry for your loss. 
My Lurcher grieved terribly for our Oldgirl when we lost her last spring. He's always been a very laid back (almost horizontal) dog that nothing ever bothered, but he started howling (really badly) when left. Even when I was just popping out for just 15 minute school runs he would start howling the second the front door shut. He was thoroughly depressed at home and only ever happy when out walking, so I reached a point where we were walking for 3-4 hours a day while the ds were at school and I was getting nothing else done.
I did a lot of research and a lot of work with him and discovered it wasn't separation anxiety, but something called isolation distress. He wasn't over attached to us - or Oldgirl - he had simply never had to be totally alone before and couldn't handle it. (Although, realistically, there was probably also a certain degree of anxiety because Oldgirl was very much the matriarch of the household, dogwise at least and he was now left as the only dog in the house with noone to lead the way and tell him what to do - she was a confident bossy girl.)
I was advised to get another dog by a couple of behaviourists, but didn't want to do so without helping him learn to cope better first, so spent three months working on gradually getting him used to being alone. Started with just a few minutes, worked on desensitising him to signals that I was about to leave and worked up to him eventually being able to be left for and hour and a half. At that point we were approached about taking on a rescue Lurcher pup and felt he had improved sufficiently enough for us to do so.
Ironically, the Lurcher pup we took on had had a very bad start and developed Separation Anxiety as a result.
He's 9 months old now and I was able to leave him for an hour and a half without him yelling for the first time yesterday! Older Lurcher grudgingly accepts loves him now, but was actually mightily unimpressed that we brought another dog into the house at first - very similar Slapntickle's boy's reaction really.
I'm glad, despite what hard/stressful work it was, that we did the training with our older Lurcher first, as it meant when the pup needed to go to the vets (which he did a lot at first) or puppy classes etc, he was able to cope with being home alone and I didn't have to worry about him.
Bearing in mind that Greyhounds are usually raised in kennels, it's very likely that your boy has never been on his own and simply doesn't have the skills to cope with it. I would recommend trying to teach him those skills first, before you leap into getting another dog, as it should be fairly obvious quite quickly whether or not he's going to be able to learn to cope.