You need a very sturdy top to the cage or find a way to put it up high enough the dog cannot get to it -- and then you still need a top to it. Dogs are great at jumping, grabbing a bite of cage with their teeth and yanking the thing down. They are good at lifting up a top with their noses. Dogs can be lightning fast when they decide the guinea pig they see would be the perfect snack or play toy, and I doubt you'd be fast enough to stop him/her.
- And got the T-shirt
Don't cover them up. Get some closet shelving (Closetmaid kind), and put lengths of it over the cage. If you can raise the cage on a table, you can interact with them easier, and the shelving will protect them from the dog.
There are inexpensive utility tables with white plastic tops that you can get. Some even fold. I have two of them and find any number of uses for them. They have adjustable legs to put them at different heights. If your cage has a sturdy bottom, you could buy two of them and put one under either end of your cage. If the cage needs even more support, you could purchase three of them. For added safety, I would lash the cage to the tables. Then use the closet shelving over the top. You can use zip ties to fasten the shelving onto the cage and to each other to end up with a top that opens and closes. Use sturdy clips to hold it closed on the end that can be raised up. In addition to this, you can fasten the cage to the wall so the dog will not push/pull it over if he jumps on it.
The dog that you describe is a powerful animal. Even with the above setup, it might try and harm your pigs. One strong paw through the bars of a cage could cause a fatal injury. If you see the dog trying to paw the cage, you need to take additional steps to protect them.
The other part of this safety set up is to get a crate for your dog and teach the dog to go in the crate when you have to leave the house. Because you just cannot trust the dog when you are not there. You may also need to crate the dog when you are handling the pigs or when they are out during a cage cleaning. If a 40 lb. dog that is all muscle (like your type of dog is) tries to go for a pig on your lap, there's no way you are going to be able to block it.
How to introduce the dog to your pigs? I would start with a small stuffed animal. Rub it through the pig cage to get the smells on it and then hold it in your lap and let the dog sniff it. Have a small string attached to it and place it on the floor so you can pull it along. Judge how the dog responds. Does it back off if you say "no?" ANY show of too much interest or aggression and I would not even dream of letting it interact with a live pig.
Also, even if you don't have a separate room for the piggies, you can put up gates or something to keep your dog out of the room with the guinea pigs while letting you be in there.