In this paper, we consider third-order linear recurrences {un}n≥0 satisfying the recurrence relation un+3=un+2+un+1+un for all n≥0 and investigate the multiplicity of its zeros. We prove that {un}n≥0 has zero-multiplicity at most 2, except for nonzero multiples of shifts of the Tribonacci sequence which has zero-multiplicity 4 when the indices are extended to all the integers.