Monkey grass (Mondo grass) and liriope are not poisonous plants. Monkey grass can be irritating to the GI tract when eaten in large quantities. Despite popular belief dogs are not actually carnivores. They are omnivores and require significant levels of carbohydrates and fiber in their diets. If your dog is out "grazing" it is most likely a sign that he isn't getting enough fiber in his diet. If he is already on a good quality food (and the saying you get what you pay for usually holds true with dog food too) then you may need to switch him to a higher fiber diet. The other option is to add some vegetables like raw carrots, green beans, romaine lettuce, broccoli (it can cause gas), etc. (avoid corn and potatoes as they are high in calories and avoid onions because they are toxic to dogs). This may help satisfy his nutritional desire for the fiber as well as the oral (tactile) enjoyment.
Because he has gotten into the habit of eating the monkey grass all the time you will probably have to do some training to prevent him from continuing to eat it. A squirt gun (if he doesn't think it is a game) to use every time on him when he goes for it will often work as long as you are persistent. You can spray a mist of bitter apple or tabasco onto the plants to make them taste bad.
One way to tell if your dog is getting enough fiber is by his bowel movements. He should ideally have two a day and they should be formed, but not hard. With more fiber in his diet he will probably drink a little more water also. There are not many other medical problems to cause a dog to eat plants, but not feeling good can chane behavior so it would be a good idea to get a physical exam, CBC, Chemistry, urinalysis, and thyroid level done to make sure there is no serious underlying problem.