The API allows you to nest AND and OR conditions in a tree like structure. For example, to implement the where condition you described above you will have to do something similiar to the below (Pseudo code.. don't expect it to compile):
qry.Where = new And
{
Conditions = new Condition[]
{
new Or
{
Conditions = new Condition[]
{
new Compare
{
LeftExpression = new FieldExpression { FieldName = "Submitted" },
Operator = Operator.Equal,
RightExpression = ...
},
new Compare
{
LeftExpression = new FieldExpression { FieldName = "AssignedTo" },
Operator = Operator.Equal,
RightExpression = ...
},
}
},
new Compare
{
LeftExpression = new FieldExpression { FieldName = "Manager" },
Operator = Operator.Equal,
RightExpression = new ConstantExpression{Value = "Some String"}
},
}
};