Rent a car from Allied Car Rental or Charlie's car rentals, where they usually have decent prices on Jeeps and convertibles, and drive east out of Isla Verde along Route 187 toward Fajardo. If you have a 4 wheel drive vehicle, you can see small sand entrances amongst dense trees that you can drive into, leading you to semi-private or private beach areas. These areas aren't great for swimming but you can certainly cool off privately or stop for a picnic.
See also "Get Around" "Rental Car" under San Juan for driving there.
