Despite my ex being bi-curious himself, he would occasionally comment that my bisexuality made him worry that I would cheat on him with a girl. He didn't trust me as much as he would have trusted a straight partner. My current boyfriend on the other hand is more than happy to have a bisexual girlfriend. It makes threesomes a little easier to arrange!
I agree that you may want to take a close look at your relationship if your lady isn't ok with you being who you are.