The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that visits as many nodes as possible during their respective time windows. A polynomial-time bicriteria approximation algorithm is presented for this problem, gaining an increased fraction of repairman visits for increased speedup of repairman motion. For speedup $s$, we find a $6γ/(s + 1)$-approximation for $s$ in the range $1 \leq s \leq 2$ and a $4γ/s$-approximation for $s$ in the range $2 \leq s \leq 4$, where $γ= 1$ on tree-shaped networks and $γ= 2 + ε$ on general metric graphs.