New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+Hankel like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured projections, which have been recently successfully applied for computing the bivariate resultant. A first baby-step/giant step approach - directly derived using known techniques on structured matrices - gives a randomized Monte Carlo algorithm for the minimal polynomial of an n×n Toeplitz or Hankel-like matrix of displacement rank alpha using O˜(n^(w-c(w)) alpha^c(w) ) arithmetic operations, where w is the exponent of matrix multiplication and c(2.373) ≈ 0.523 for the best known value of w. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in O˜(n^(2−1/w)) operations when the displacement rank is considered constant. Previous algorithms required 2 operations while the exponents presented here are respectively less than 1.86 and 1.58 with the best known estimate for w.