The paper considers the heat kernel KΩ(t, x, y) of the operator – Δ on a proper Euclidean domain Ω, with Dirichlet boundary conditions. A general pointwise lower bound for KΩ, which is valid for t larger than a suitable t0(x,y), is proved (the short-time behaviour being well understood). The resulting non-Gaussian bounds describe simultaneously both the case of bounded domains and the case, modelled on the half-space example, of domains which satisfy a twisted infinite internal cone condition. Bounds for the Green's function are given as well.