This paper is concerned with the problem of stability of linear time-delay systems. Firstly, a new approach, i.e. a delay decomposition approach, is proposed to deal with the problem. The idea of the approach is that the delay interval is uniformly divided into N segments with N a positive integer, and a proper Lyapunov-Krasovskii functional is chosen with different weighted matrices corresponding to different segments in the Lyapunov-Krasovskii functional. Secondly, based on the delay decomposition approach, some new delay-dependent stability criteria for linear time-delay systems are derived. These criteria are much less conservative and include some existing results as their special cases. Numerical examples show that significant improvement using the delay decomposition approach is achieved over some existing methods even for coarse delay decomposition. For fine delay decomposition, the delay limit for stability can be approached.