Grass type Pokemon already use a few Ground type moves. (like Dig, Earthquake, Bulldoze)
Case-by-case basis, actually. Ground-type moves are not very common on Grass-types at all. Any ones that do are either large or have some connotation with the ground anyway, or the move is easy enough to execute for them (Dig).
Just out of interest, I did a search of what Grass-types get those moves, and I got this:
Bulldoze: Abomasnow, Arceus (Meadow Plate), Cradily, Ferrothorn, Meganium, Sceptile, Tangrowth, Torterra, Tropius, Venusaur
Dig: Grovyle, Leafeon, Nuzleaf, Pansage, Sceptile, Seedot, Shiftry, Simisage, Treecko
Earthquake: Abomasnow, Arceus (Meadow Plate), Cradily, Meganium, Sceptile, Tangrowth, Torterra, Tropius, Venusaur
All of the above are learned via TM, save for Torterra learning Earthquake (at level 32). And this list was a lot smaller than I thought it would be. The Bulldoze and Earthquake lists are identical, except that Ferrothorn cannot learn Earthquake.
While it's true that Grass-types do get Ground-type moves, sometimes, it is not because of any innate connection with the ground they might have. For Bulldoze and Earthquake, it's generally sheer size (being able to create such tremors), and for Dig, it's just the ability to dig one's self into the ground. (Seedot is the odd one, but it's based on an acorn, which must be planted in the ground to grow, for it is a seed.)
...which means there's nearly no correlation between type and moves learned - in fact, moves learned tend to be based on other characteristics, usually - mostly physical, but others possibly mental (like Gliscor learning Power Trick).