At the most basic level you can't - given an arbitrary object o (and the question implies you're looking for generic solutions) you have no way to determine which field is a primary key.
So you step up a level - by adding some constraints i.e. that you will inform your tools what the primary key field is (or fields are) and hence enable use of a generic method.
So, you could explicitly specify the PK field (name) to the code that does the clone (I assume that you're using reflection to avoid explicitly copying all the fields). You could identify the PK by using annotation of some sort on the classes being cloned and have the clone code exclude properties with the relevant annotation (the annotation implies that the field won't be cloned). There may be other methods
You mention Linq - are you using a specific bit of Linq ?
Beyond that there's not a lot one can suggest without more details - ah but the question is tagged with Linq to SQL (which I missed) ok...
There's nothing obvious in a Linq to SQL class that will help - nor with the "table" but a quick look at the generated code in .designer.cs shows that a key field has annotations similar to the following (taken from a set of classes I have to hand):
[Column(Storage="_ID", AutoSync=AutoSync.OnInsert, DbType="Int NOT NULL IDENTITY", IsPrimaryKey=true, IsDbGenerated=true)]
Therefore when you're do your reflection on the class to enumerate the properties to copy you'll want to look for the column and the "IsPrimaryKey" property within the column - unfortunately the details of how to do that are some distance outside my comfort zone!