Since you say that it was overall a positive experience, I don't think there is anything wrong with mentioning your previous interviews, especially if you had positive feedback. You can also acknowledge the possible drawbacks (i.e. the fact that the interviews were unsuccessful):
"I was drawn to this position after being considered for a previous
role in your company. Although I was unsuccessful then, I am better
suited to this opening because [etc.] and have since improved by
accomplishing [amazing stuff]. My previous interactions with the
company have been extremely positive and the workplace atmosphere has
only increased my interest."
That way, you're building on any positive memories other staff might have of you, acknowledging your strong interest in the company (you're bothering to reapply), essentially saying that you've got past the "first date" stage with their company, you know the basics and you still want to work there. You're also saying that now you're in a better position to meet their needs, and saying how you've improved since then. Generally, companies do want the good candidates to reapply!
As you say, there is a risk. However, given that you were beaten to other role by a candidate who had experience that you couldn't possibly have, as a graduate, they may be a bit more sympathetic, especially given your motivation to work there. If you are especially worried, why not get in touch with one of your previous interviewers and ask what they think? They'll be able to give you more specific advice about the department you're applying to and the likely reaction of the people interviewing - and who knows, they may even put in a good word for you!