Define an infinite lower triangular matrix D(e,h) = [d_n,k]_{n,k ≥ 0} by the recurrence d_0,0 = d_1,0 = d_1,1 = 1, d_n,k = d_n-1,k-1 + ed_n-1,k + hd_n-2,k-1, where e, h are both nonnegative and d_n,k = 0 unless n ≥ k ≥ 0. We call D(e, h) the Delannoy-like triangle. The entries d_n,k count lattice paths from (0, 0) to (n - k, k) using the steps (0, 1), (1, 0) and (1, 1) with assigned weights 1, e, and h. Some well-known combinatorial triangles are such matrices, including the Pascal triangle D(1, 0), the Fibonacci triangle D(0, 1), and the Delannoy triangle D(1, 1). Futhermore, the Schröder triangle and Catalan triangle also arise as inverses of Delannoy-like triangles. Here we investigate the total positivity of Delannoy-like triangles. In addition, we show that each row and diagonal of Delannoy-like triangles are all PF sequences.