Be honest. They will appreciate our honest and it will be easier for both of you to move on from the break up. It is best to tell them it just isn't working for you anymore and you think the relationship should end. Chances are they might agree and the break up will become mutual.
Tell them that they are a phenomenal person with so many amazing traits that will be valued more by another person. Tell them you aren't the one that can appreciate what they offer in a relationship. This speech will keep a balance between you.
Become intellectually boring. Virgos love to talk about the world, news, politics, travel and tell stories. Act dumb and they will have you out the door in an hour. They need constant intellectual stimulation and if you do not provide it, it will be the end.
Tell them to spend more time with you instead of working. Virgos value their career and love to get themselves into all sorts of challenges and projects. If you act needy and demand loads of attention, they will surely start to feel threatened.
Act vague and mysterious. The more you hide things, the more they will begin to mistrust you. They value trust and honest at all times, and if you become mysterious then they will become suspicious. This behavior in turn might break you both up.