Yes, there were several.

One was called tesserae. A tessera in Latin is just any small cube or square token -- so the pieces of a mosaic are tesserae. It was also one word for dice however (shared with alea). Roman dice were just like modern dice. These are from the Met, but there are thousands of ancient examples in existence. It is the type of thing that gets carried away from excavations by naughty grad students. The origins of the game are Greek, and the object was to make certain throws, not unlike modern craps. There was also sometimes a board involved, like this famous one scratched into the steps of the Basilica Iulia in the Forum Romanum. I don't know the specifics of how the game worked, or how the board worked, but there were usually three dice used. The best throw was, of course, three sixes, as Aeschylos makes clear in Agamemnon :

Watchman: Fire gleaming in the night! What a welcome sight! Light of a new day— you’ll bring on many dancing choruses right here in Argos, celebrations
of this joyful news. It’s over! It’s over! I must call out to wake the queen, Clytaemnestra, Agamemnon’s wife, to get her out of bed, so she can raise a shout of joy as soon as possible inside the palace, welcoming this fire— if indeed the city of Troy’s fallen, as this signal fire seems to indicate.
For my part, I’ll start things off by dancing, treating my king’s good fortune as my own.
I’ve had a lucky dice roll, triple six, thanks to this fiery signal . . . . (transl. Ian Johnston)

Another, similar game was tali ('bones'), which used animal knucklebones instead of dice. They only had four numbered sides instead of six, and again the object was to throw certain combinations of numbers. The knucklebones of exotic animals were premium grade (like having a meteorite 20-sided these days). Some were super fancy. Tali was often associated with kids, women, and old men.

Technically, games of chance involving gambling were forbidden in Republican and early Imperial Rome and was certainly not something a senator should be doing (so Cicero calls Catilina an aleator ('gambler')(In. Cat. 2.10).