With a mean-field treatment, the spin tunneling of spin-1 Bose-Einstein condensates in a linearly sweeping magnetic field is investigated. We focus on a typical alkali Bose atoms 87Rb condensate and study its tunneling dynamics under different sweeping rates of external magnetic field. In the adiabatic (i.e., slowly sweeping) and sudden (i.e., fast sweeping) limits, no tunneling is observed. In the case of moderate sweeping rates, the tunneling dynamics is found to be very sensitive to the sweeping rate, showing a chaotic-like tunneling regime. By magnifying the apparently chaotic regime, however, we find interestingly that the plot becomes resolvable at a resolution of 10-8T/s where the tunneling probability with respect to the sweeping rate shows a regular pseudo-periodic pattern. Moreover, a conserved quantity standing for the magnetization in experiments is found which can dramatically affect the above picture of the spin tunneling. Theoretically, we give a reasonable interpretation to the above findings.