We consider the fluctuation of linear eigenvalue statistics of random band n×n matrices whose entries have the form Mij = b −1/2 u 1/2 (|i − j|)wij with i.i.d. wij possessing the (4 + ε)th moment, where the function u has a finite support [−C * , C * ], so that M has only 2C * b+1 nonzero diagonals. The parameter b (called the bandwidth) is assumed to grow with n in a way that b/n → 0. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers [8] and [11], where CLT was proven under the assumption n >> b >> n 1/2 . Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.