Ideally (L’chatchila), there is a Mitzvah to perform Zimun over a cup of wine, as it is written: “I will lift the cup of salvation and call upon the name of the Lord” (Siman 182, Se’if 7 and Mishna Berurah ibid.). However, when there is no wine available in the house, we are lenient and may perform the Zimun without a cup.

There is a dispute among the authorities as to whether Birchas HaMazon requires a cup even when eating alone (Yachid); however, the common custom is not to use a cup when eating alone (Siman 182 and Mishna Berurah ibid.). Even when three people eat together, the general practice is to be lenient and recite the blessings without a cup (ibid. and Kaf HaChaim).

On the other hand, I found in Ohr L’Tzion (Part 46) that it is proper to be stringent when three people eat together and recite the blessings over a cup.