The activation method was used to measure cross section values from threshold energies up to 20 MeV for the reactions 64Zn(n,p)64Cu, 68Zn(n,α)65Ni, and 70Zn(n,2n)69Znm. The neutron flux at energies below 5.3 MeV was determined with a calibrated neutron long counter while at higher energies measurements were made relative to the known cross section for the 32S(n,p)32P reaction. Effective cross sections for a fission-neutron spectrum calculated from measured excitation curves are 42.3 ± 2.1 mb for the (n,p) reaction, 0.035 ± 0.003 mb for the (n,α) reaction, and 0.45 ± 0.04 mb for the (n,2n) reaction. Factors influencing precise measurements of fast-neutron cross sections are discussed in detail.