2003 Pontiac Grand Prix Radiator Gunk

Brenda, GM 3.1 & 3.4 litre engine are known for the intake gaskets leaking. If your intake gasket is begining to leak internally the engine oil pressure is higher than the engine coolant system pressure, alowing engine oil to get into the coolant system and would cause the brown gunk (oil) in your coolant system. You should also notice symptoms like loss off coolant or oil.
You can check this by looking at the ends of the engine where the intake meets the engine block for coolant or oil pools (for external leaks) and pressure check the cooling system (for internal leaks).
Let me know if this helps.
Dan

GM recalled many 3.1, 3.4 and 3.8 engines because of intake gasket leaks. Check with GM, they should replace it free and drain and flush the cooling system.

Mmprince4000 is correct, they did recall some. I did check before my first post, I did not find any recalls of the intake or gasket for your model year, but there are technical service bulletins for the intake. Revised bulletin number 03-06-01-010C for the 3.1 and 04-06-01-017 for the 3.8 litre engines.
Technical service bulletins give information of an updated part or repair proceedure, but are not covered under warranty past the original factory warranty. But as mmprince4000 said, you should definately check with the dealer first.
Also, the service manager at a dealership does have the authority to do repairs under warranty past the expiration date but runs the risk on the manufacturer not reimbursing the dealership for the repairs. With known conditions like this they may offer a 50/50 split of the repair cost to the customer as well.
I would recommend contacting your local Pontiac (GM) Dealer and seeing if they would be able to help you out with this.
Ask to speak to the service manager. Settting up a good repore with the service manager is important. They are more willing to help out regular customers and people that approach them with a positive attitude.
Hope this helps, let me know.
Dan

As Dan mentioned, GM was sued over the intake gasket and Dex-Cool coolant and lost.

They did make an effort to replace the intake on many cars, but it turned out the replacement gaskets were also bad.

Talk to dealer about it, and see if you can get it done at cost or free. This repair usually runs $700 at a dealer.