We develop a systematic algorithm to construct, classify and study exact solutions of type IIA/B supergravity which are time-dependent and homogeneous and, hence, represent candidate cosmological backgrounds. Using the formalism of solvable Lie algebras to represent the geometry of non-compact coset manifolds U/H we are able to reduce the supergravity field equations to the geodesic equations in U/H and rephrase these latter in a completely algebraic setup by means of the so-called Nomizu operator representation of covariant derivatives in solvable group manifolds. In this way a systematic method of integration of supergravity equations is provided. We show how the possible D=3 solutions are classified by non-compact subalgebras G⊂E8(8) and their ten-dimensional physical interpretation (oxidation) depends on the classification of the different embeddings G E8(8). We give some preliminary examples of explicit solutions based on the simplest choice G=A2. We also show how, upon oxidation, these solutions provide a smooth and exact realization of the bouncing phenomenon on Weyl chamber walls envisaged by the cosmological billiards of Damour et al. We also show how this physical phenomenon is triggered by the presence of Euclidean D-branes possibly interpretable at the microscopic level as S-branes. We outline how our analysis could be extended to a wider setup where, by further reducing to D=2,1, more general backgrounds could be constructed applying our method to the infinite algebras E9,10.