A new configurational-bias Monte Carlo (CBMC) method is developed for lattice polymer simulation. In contrast to usual approaches, no particular structure is assumed a priori. Instead, any regular lattice (whose sites are connected to their neighbors by the same vectors) can be used. The method includes new features such as an algorithm to distinguish all conformations of short ideal chains in regular lattices and a simple selection scheme that can improve general CBMC methods. The method is suitable for simulating molecules with branches and/or closed cycles, allowing the study of polymers with complex, well-defined architectures. It was applied to simulate the conformational and thermal properties of multicyclic molecules in dilute solution. The results indicate systematic changes in the behavior of topologically constrained polymers, such as a shrinking of the radius of gyration, a slight reduction in scaling exponents, and an increase of the globule-coil transition temperatures.