In this paper, we focus on fast solvers with linearithmic complexity in space for high-dimensional time-fractional subdiffusion equations. Firstly, we present two alternating direction implicit (ADI) finite difference schemes for the two-dimensional time-fractional subdiffusion equation that are convergent of order (1 + β) in time, where β (0 < β < 1) is the fractional order. Secondly, we develop two finite difference schemes which admit fast solvers without applying ADI techniques for two-dimensional time-fractional subdiffusion. Lastly, we extend these fast solvers to three-dimensional time-fractional subdiffusion. All the non-ADI difference methods are unconditionally stable and convergent with order two in time and order two or four in space. We also present several numerical experiments to verify the theoretical results.