We consider message broadcasting in networks that have almost tree topology. The source node of the input network has a single message which has to be broadcasted to all nodes of the network. In every time unit each node that has already received the message can send it to one of its neighbors. A broadcasting scheme prescribes in which time unit a given node should send a message to which neighbor. It is minimum if it achieves the smallest possible time for broadcasting the message from the source to all nodes. We give the following algorithms to construct a minimum broadcasting scheme for different types of weakly cyclic networks: A linear-time algorithm for networks whose cycles are node-disjoint and in which any simple path intersects at most O(1) cycles. An O(nlogn)-time algorithm for networks whose cycles are edge-disjoint and in which a node can belong to at most O(1) cycles. An O(nk log n) -time algorithm for networks whose each edge-biconnected component is convertible to a tree by removal of at most k edges. We also present an O(n4k+s)-time algorithm for constructing a minimum broadcasting scheme for partial k-trees.