Many people have the custom of saying grace after meals on Shabbat while holding a full cup of wine in their right hand. At the end of bensching, they say the blessing on wine and drink. Some sources say that one should do this after every meal. The predominant view is that while it is a good thing to bensch, with a cup of wine, it is not necessary. There are even those who said that if no wine is available, then it is better not to eat bread, rather than to have to bensch without a cup of wine (Talmud Bavli 107a).
I do not know of anyone who follows this opinion today. If you don't have wine, then you can use beer or any drink classed as hemer d'medinah (something that one would serve guests, but not water).
Bensching on a cup of wine is in line with many other things in Judaism that are done with wine - eg kiddush, a brit and a wedding ceremony, even when the ritual and the drink have no intrinsic connection. "Wine rejoices the people's hearts," (Psalms 104).
