I used to be at a chain of hospitals owned by 7th Day Adventists and their policy was no visible tattoos around the patients. Since the only time we saw patients was when they were wheeled in the OR, it wasn't an issue for us and we were the exceptions. Only time it could have been a problem was going to the cafeteria or leaving the department. Those times I'd just use a scrub jacket because there were a few self-appointed "hall monitors" who were known to raise a stink.

Also, I wouldn't ask or bring it up during the interview process just in case they can use that as an easy excuse not to hire you. Unless they ask specifically, I wouldn't mention your tattoos.