Novel linear algorithms are proposed in this paper for estimating time-varying FIR systems, without resorting to higher order statistics. The proposed methods are applicable to systems where each time-varying tap coefficient can be described (with respect to time) as a linear combination of a finite number of basis functions. Examples of such channels include almost periodically varying ones (Fourier Series description) or channels locally modeled by a truncated Taylor series or by a wavelet expansion. It is shown that the estimation of the expansion parameters is equivalent to estimating the second-order parameters of an unobservable FIR single-input-many-output (SIMO) process, which are directly computed (under some assumptions) from the observation data. By exploiting this equivalence, a number of different blind subspace methods are applicable, which have been originally developed in the context of time-invariant SIMO systems. Identifiability issues are investigated, and some illustrative simulations are presented.