The effect of dephasing and relaxation on electron transfer in bridged molecular systems is investigated using a simple molecular model. The interaction between the molecular system and the thermal environment is described on the level of the Redfield theory, modified when needed for the description of steady-state situations. Noting that transient as well as steady-state measurements are possible in such system, we discuss the relationship between the rates obtained from these different types of experiments and, in particular, the conditions under which these rates are the same. Also, a formal relation between the steady-state rate for electron transfer across a molecular bridge and the conductance of this bridge when placed between two metal contacts is established. The effect of dephasing and relaxation on the electron transfer is investigated, and new observations are made with regard to the transition from the superexchange to the thermal (hopping through bridge) regime of the transfer process. In particular, the rate is temperature-independent in the superexchange regime, and its dependence on the bridge length (N) is exponential, exp(-beta N). The rate behaves like (alpha(1) + alpha(2)N)(-1) exp(-Delta E/k(B)T) beyond a crossover value of N, where Delta E is the energy gap between the donor/acceptor and the bridge levels, and where alpha(1) and alpha(2) are characteristic times for activation onto the bridge and diffusion in the bridge, respectively. We find that, in typical cases, alpha(1) >> alpha(2), and therefore, a region of very weak N dependence is expected before the Ohmic behavior, N-1, is established for large enough N. In addition, a relatively weak exponential dependence, exp(-alpha N), is expected for long bridges if competing processes capture electrons away from the bridge sites. Finally, we consider ways to distinguish experimentally between the thermal and the tunneling routes.