my last open heart surgery was in 1993 and I was 13, so I probably wasn't even wearing a bra yet. But, I've been in the hospital for many other surgeries, heart related and non heart related. Over the years, I have learned to bring underwear and a couple pairs of yoga/comfortable pants or shorts. I like to walk to halls in hospitals, and the gowns do not cover your backside when you are up walking. However, your breasts should be covered by the gown when you are in the recovery room. So, you shouldn't have to worry about that. you may not be able to wear a bra right away, but I don't know from experience.
Also, as a previous poster said, they will be opening your gown to check your incision daily, so you will get used to it. Over the years, I've had 5 heart catherizations and 1 heart ablation (3 years ago) when they had to go in through my groin. I was not allowed to wear underwear for those procedures. You get used to having no modesty in the operating room..haha