"Cina" is pretty standard.
They were probably using "masa de seară" (literally the evening meal), which is also very common.
Another common thing is to say "de seară"/"de prânz"/"de dimineață" with the meaning of "for dinner"/"for lunch"/"for breakfast", effectively using times of day to refer to meals.
What are we eating for dinner? - Ce mâncăm de seară?
Yes, you can use them without the preposition. "Nu dorm noaptea" and "Nu mănânc dimineața" would be grammatically OK in Romanian, but I think they would be used in general cases like "I'm not sleeping nights because I usually work at night." or "I'm not eating in the morning because I'm on a weird diet."
This correlates to a question I was going to ask. The answer I believe is: no
seara is how one would say, "the evening" while seară is just "evening"
In this use, the phrase translates as "Dinner is in the evening."
The 'ă' becomes 'a' [think of masă and masa: the difference between 'table' and 'the table']
I was going to ask if one could translate it as "Dinner is in evening." -- but upon seeing your question, I believe this is the answer now. I am still learning so if anyone can confirm or deny this please help us out. Thanks!