Romania is one of the most religious countries in Europe, and the Orthodox church is omnipresent. You will certainly want to visit some churches and monasteries for their beauty and history, but why not take the chance to experience an Orthodox mass? The congregation is usually standing and it is perfectly normal to show up only briefly during the mass so you can come and go at your leisure without disturbing anyone. Show up at any church on sunday morning, stand quietly in the back and observe. Be suitably dressed, see the section "Respect".
You will experience bible readings, prayers and other rituals accompanied by a short sermon explaining the text. You are not likely to understand much, but you can notice the varying levels of involvement among church-goers, visible in how long and where people stay at the mass, and how often they sign themselves with the cross, or even genuflect. Organized congregation singing is not common but is conducted by a choir with each church-goer joining when he feels like. The choir singing can be captivating, the quality usually reflects the importance of the church.
The altar has sections with doors that open and close depending on the church season. You will also see candles sold, they are lit in or by the church in separate trays for the souls of either dead or living people. Try to find out about special holidays and rituals, perhaps the distribution of holy water by the truckload at the baptism of Christ Boboteaza or midnight masses at Christmas or Easter the Orthodox Easter may be off by one week compared to the Western. Weddings are often saturdays, the ritual is very colorful and interesting.