Let \(A=(a_{nk})\) be an infinite matrix of complex numbers and \(x=\{x_k\}\) a sequence of complex numbers. The sequence \(\{A_n(x)\}\) defined by \(A_n(x)= \sum^\infty_{k =0} a_{nk}x_k\) is called the \(A\)-transform of \(x\) if it converges for every \(n\). The concepts of \(A\)-summable sequence, almost conservative matrix and almost regular matrix were introduced by J. P. King [Almost summable sequences, Proc. Am. Math. Soc. 17, 1219-1225 (1966; Zbl 0151.05701)]. A lacunary sequence is an increasing sequence \(\theta= (k_r)\) such that \(k_0=0\) and \(k_r-k_{r-1} \to\infty\) as \(r\to\infty\). For a lacunary sequence \(\theta\) the author defines \(\theta\)-almost convergence and \(\theta\)-almost \(A\)-summability of \(x\), and \(\theta\)-almost conservativeness and \(\theta\)-almost regularity of \(A\), and proves some theorems giving characterizations of these concepts.