Detection of the organic matter on Mars is one of the main goals of the future Martian landing missions. Yet, the degradation of organic molecules by cosmic ray irradiation on Mars is often ignored. We calculate the radiation dose accumulation rates from solar and galactic cosmic rays at various depths in the shallow Martian subsurface. We demonstrate that a 1-billion-year outcrop on Mars accumulates the dosage of ~500 MGy in the top 0-2 cm and ~50 MGy at 5-10 cm depths. We show that the preservation of ancient complex organic molecules in the shallow (~10 cm depth) subsurface of rocks could be highly problematic if the exposure age of a geologic outcrop would exceed 300 Myr. We demonstrate that more simple organic molecules with masses ~100 amu should have a good chance to survive in the shallow subsurface. Implications to the sampling strategy for the oncoming Martian missions are discussed.