This paper presents new measurements of the carbon and beryllium neutron total cross section in the energy range of 24 to 950 keV. The measurements were done using a pulsed neutron source driven by an electron LINAC. The neutron beam passed through a 30-cm-thick iron filter, which results in neutron transmission only in energies where resonance scattering and potential interference exist. The neutron filter removes most of the neutrons at other energies and significantly attenuates the gamma background resulting in 20 energy windows and a high signal-to-background ratio. The filtered beam was used for transmission measurements through graphite that results in ~1% accurate total cross sections that are in excellent agreement with current evaluations. The carbon measurement provides a verification of the accuracy of the filtered beam method. Measurements of three samples of different thicknesses of beryllium resulted in accurate total cross-section values that agree with one previous measurement and show discrepancies from current evaluations. The high accuracy of the new measurements can be used for improvement of future total cross-section evaluations of beryllium.