I recently used 3% Peroxide solution to lighten my hair, it looks great however anytime my hair gets a little greasy, it is really obvious now.

What can I do to avoid this?

Use hair dye.

When peroxide reacts with hair, it lightens the melanin in the hair. Normally, this turns the hair reddish or orange if it's lighter hair. Hair dye contains dyes that are deposited (by the action of the peroxide) into the hair shaft. These dyes counteract the red/orange from what is left of the melanin.