We propose a single source reliable broadcasting algorithm for linear grid-based networks where a message is guaranteed to be delivered to all the nodes of the network. The nodes are mobile and can move from one grid point to another. The solution does not require the nodes to know the network size or its diameter. The only information a node has is its identity and its position. On average, only a subset of nodes transmit and they transmit only once to achieve reliable broadcast. The protocol is contention-free and energy-efficient. We show that reliable broadcast can be achieved in O(Dlog n) time-slots despite node mobility, where D is the diameter of the network and n the number of nodes.