In this paper some aspects of approximation of periodic Besov classes \(B_{p,\theta}^r= B_{p,\theta}^r(\pi_d)\), \(\pi_d=\prod_{j=1}^d[-\pi,\pi]\), \(1\leq p\leq \infty\), by linear methods, and of their best approximation by trigonometric polynomials with numbers \(k=(k_1,\dots,k_d)\) of harmonics \(e^{i(k,x)}\) from the stairwise hyperbolic cross in the space \(L_\infty(\pi_d)\) are considered. The behaviour of the norms of the sequence of the linear operators is investigated which guarantee the same order of approximation of the classes \(B_{p,\theta}^r\) in the uniform metric as their best approximations. It is shown that approximation of the classes \(B_{p,\theta}^r\), \(1<\theta<\infty\), \(1\leq p\leq 2\), by trigonometric polynomials via the linear methods in the uniform metric does not realize the order estimates of the corresponding best approximations.