A local search algorithm solving an NP-complete optimization problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satisfiability problem, the heuristic of focusing the local moves on the currently unsatisfied clauses is known to be very effective: the time to solution has been observed to grow only linearly in the number of variables, for a given clauses-to-variables ratio α sufficiently far below the critical satisfiability threshold α_c≈4.27. We present numerical results on the behaviour of three focused local search algorithms for this problem, considering in particular the characteristics of a focused variant of simple Metropolis dynamics. We estimate the optimal value for the 'temperature' parameter η for this algorithm, such that its linear time regime extends as close to α_c as possible. Similar parameter optimization is performed also for the well-known WalkSAT algorithm and for the less studied, but very well performing focused record-to-record travel method. We observe that with an appropriate choice of parameters, the linear time regime for each of these algorithms seems to extend well into ratios α>4.2—much further than has so far been generally assumed. We discuss the statistics of solution times for the algorithms, relate their performance to the process of 'whitening', and present some conjectures on the shape of their computational phase diagrams.