A history of family members with heart defects (ASD, VSD, tetralogy of fallot) will prompt us to offer you a fetal echocardiogram at 20 weeks, because heart defects can have a hereditary basis (they are multifactorial- they have a genetic and environmental basis).
Chromosomal problems such as Down syndrome and the age at which the mother was when she had the baby are important. Mental retardation in a family member, especially in boys or with multiple relatives will alert us to look for a fragile X mutation which can be picked up by a blood test.
Muscular dystrophies, dwarfism, cystic fibrosis and other medical issues can be hereditary and can be screened for (depending on the disease preconceptually or postconception through invasive fetal testing such as CVS or amnioscentesis).
If there is a disease, common or uncommon, that runs in your family, it is imperative to speak to your Ob/Gyn about it preconceptually. Sometimes we even send you for genetic counseling and testing before you get pregnant.
Ethnicity of both parents is also important. There is now Jewish genetic screening available for eleven hereditary Jewish diseases (it's not just Tay Sachs anymore). African-American patients should be screened for sickle cell disease, mediterranean and asian patients should be screened for thalassemias.